arXiv:2106.15656v1 [astro-ph.CO] 29 Jun 2021

Draft version July 1, 2021

Typeset using LATEX modern style in AASTeX63

Measurements of the Hubble Constant: Tensions in Perspective∗

Wendy L. Freedman1

1Department of Astronomy & Astrophysics & Kavli Institute for Cosmological Physics, Universityof Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA

(Accepted to the Astrophysical Journal, June 23, 2021)

ABSTRACT

Measurement of the distances to nearby galaxies have improved rapidly in recent

decades. The ever-present challenge is to reduce systematic effects, especially as

greater distances are probed, and the uncertainties become larger. In this paper,

we combine several recent calibrations of the Tip of the Red Giant Branch (TRGB)

method. These calibrations are internally self-consistent at the 1% level. New Gaia

Early Data Release 3 (EDR3) data provide an additional consistency check, at a

(lower) 5% level of accuracy, a result of the well-documented Gaia angular covariance

bias. The updated TRGB calibration applied to a distant sample of Type Ia super-

novae from the Carnegie Supernova Project results in a value of the Hubble constant

of H0 = 69.8 ± 0.6 (stat) ± 1.6 (sys) km s−1 Mpc−1. No statistically significant dif-

ference is found between the value of H0 based on the TRGB and that determined

from measurements of the cosmic microwave background. The TRGB results are also

consistent to within 2σ with the SHoES and Spitzer plus HST Key Project Cepheid

calibrations. The TRGB results alone do not demand additional new physics beyond

the standard (ΛCDM) cosmological model. They have the advantage of simplicity of

the underlying physics (the core He flash) and small systematic uncertainties (from

extinction, metallicity and crowding). Finally, the strengths and weaknesses of both

the TRGB and Cepheids are reviewed, and prospects for addressing the current dis-

crepancy with future Gaia, HST and JWST observations are discussed. Resolving

this discrepancy is essential for ascertaining if the claimed tension in H0 between the

locally-measured and the CMB-inferred value is physically motivated.

[email protected]

∗ Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the SpaceTelescope Science Institute, which is operated by the Association of Universities for Research in As-tronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with programs#13472 #13691, #9477 and #10399.

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http://orcid.org/0000-0003-3431-9135

mailto: [email protected]

2 Freedman

Keywords: galaxies: distances and redshifts – cosmology: distance scale – cosmology:

cosmological parameters – cosmology: theory – cosmology: early universe

– stars: low-mass – stars: Population II –

1. INTRODUCTION

Over the last decade, the unprecedented increase in accuracy obtained by a broad

range of independent cosmological experiments and observations has provided striking

and compelling support for our current standard Λ Cold Dark Matter (ΛCDM) model.

This concordance cosmology has been remarkably successful in explaining an even

wider range of observations, from the exquisite precision in recent measurements of

fluctuations in the temperature and polarization of the cosmic microwave background

(CMB) radiation (Planck Collaboration et al. 2020; Aiola et al. 2020) to observations

of large-scale structure and matter fluctuations in the universe (e.g., baryon acoustic

oscillations (BAO), Macaulay et al. 2019).

However, as the accuracy of both the observations and the tests of ΛCDM have

improved, a number of discrepancies have been noted. The most apparently sig-

nificant of these is the claim of a tension between competing values of the Hubble

constant (H0), where the discrepancy is currently estimated to be at the 5 to 6 sigma

level (Riess et al. 2021; Di Valentino et al. 2021) between the local values of H0 and

those derived from models of the CMB.1 This claimed tension suggests that the uni-

verse at present is expanding about 8% faster than predicted assuming the ΛCDM

model, which, if confirmed, could provide evidence for cracks in the standard model,

offering the exciting opportunity for discovering new physics. Confirming the reality

of the H0 tension could have significant consequences for both fundamental physics

and modern cosmology.2 The implications of an accurate value of H0 are of interest,

however, independently of how the tension is ultimately resolved: providing inde-

pendent confirmation of the standard cosmological model would also be a critical

result.

As apparent fissures in the standard model have been emerging, there are also

indications that there may be cracks that need attention in the local distance scale as

well. For example, the Tip of the Red Giant Branch (TRGB) method and the Cepheid

distance scale result in differing values of H0 = 69.6 ± 1.9 km/sec/Mpc, (Freedman

et al. 2019, 2020, hereafter, F19, F20) for the TRGB and 73.2 ± 1.3 (Riess et al. 2021,

hereafter, R21) for the Cepheids. This divergence raises the question of whether the

1 As noted by Feeney et al. (2018), the true tension between the Planck and the SHoES results dependson accurate knowledge of the tails of the likelihoods of the two distributions, rather than assumingthem to be Gaussian. The significance of the current tension also depends on the assumption thatall sources of uncertainty have been recognized and accounted for.

2 For a different perspective on the H0 tension, see the recent review by Linder (2021).

TRGB Calibration Update 3

purported tension is instead being driven by yet-to-be-revealed systematic errors in

the local Cepheid data rather than in the cosmological models.

A number of measurements of H0 calibrated locally (referred to as late-time es-

timates) exhibit reasonable agreement to within their quoted uncertainties, generally

falling in the range of 70-76 km s−1 Mpc−1 (Freedman et al. 2012; Riess et al. 2016,

2019; Huang et al. 2020; Kourkchi et al. 2020; Reid et al. 2019; Freedman et al.

2019, 2020; Pesce et al. 2020; Khetan et al. 2021; Blakeslee et al. 2021). In contrast,

(early-time) estimates of H0 based on measurements of fluctuations in the tempera-

ture and polarization of the cosmic microwave background (CMB) from Planck and

ACT+WMAP (Planck Collaboration et al. 2020; Aiola et al. 2020) consistently yield

lower values of H0 = 67.4 ±0.5 and 67.6 ± 1.1 km s−1 Mpc−1, respectively, both

adopting the current standard ΛCDM model. Measurements of fluctuations in the

matter density or baryon acoustic oscillations (e.g., Aubourg et al. 2015; Macaulay

et al. 2019) also result in similar (low) values, if the absolute scale is set by the

sound horizon measurement from the CMB or by Big Bang nucleosynthesis (BBN)

constraints, also based on sound horizon physics.

High values of H0 were initially obtained from time-delay measurements of

strong gravitational lensing (Suyu et al. 2017; Wong et al. 2020), with H0 = 73+1.7−1.8

km s−1 Mpc−1, apparently consistent with the Cepheid measurements. However, re-

cent detailed consideration of the assumptions in the modeling of the lens mass dis-

tribution (Birrer et al. 2020; Birrer & Treu 2020) leads to a much lower value of

the Hubble constant, as well as a significantly larger value of the uncertainty, H0 =

67.4+4.1−3.2 km s−1 Mpc−1, currently consistent with the CMB and TRGB measurements.

The debate over the value of the Hubble constant is clearly not yet over. And

with the high precision of current CMB measurements, the requirement for greater

accuracy in the local value of H0 has grown substantially. Given the importance of

this question for fundamental physics and for cosmology, and given the history of

H0, and the century-long effort to address a multiplicity of systematic effects, it is

essential that rigorous tests be undertaken to investigate the possibility that remaining

(potentially unknown) systematic errors are responsible for driving the controversy.

The TRGB method has emerged as one of the most precise and accurate means

of measuring distances in the local universe. The TRGB is an excellent standard

candle, as an unambiguous signpost of the core helium-flash luminosity at the end

phase of red giant branch (RGB) evolution for low-mass stars (e.g., Lee et al. 1993;

Rizzi et al. 2007; Salaris et al. 2002; Madore et al. 2009; Freedman et al. 2019; Jang

et al. 2021). Empirically, observed color-magnitude diagrams of the halos of nearby

galaxies reveal a sharp discontinuity at a well-defined luminosity.

4 Freedman

In F19 we presented a determination of H0 based on TRGB distances to 15

galaxies that were hosts to 18 Type Ia supernovae (SNe Ia). I-band TRGB distances

were measured using HST Advanced Camera for Surveys (ACS) data targeting the

halo regions of nearby galaxies, and then applied to a sample of 99 significantly

more distant SNe Ia (out to z = 0.08) that were observed as part of the Carnegie

Supernova Project, and published in Krisciunas et al. (2017). This TRGB calibration

was updated slightly in F20, yielding a value of H0 = 69.6 ± 0.8 (stat) ± 1.7 (sys)

km s−1 Mpc−1. To date, the TRGB is the only method with comparable numbers

of galaxies in its calibration relative to Cepheids; the H0 calibration of Riess et al.

(2016, 2019, hereafter R16, R19) is based on the Cepheid distances to 19 galaxies.

Ten of the galaxies in the F19 and F20 TRGB sample also have independent Cepheid

distances, an order of magnitude greater number than for Miras (Huang et al. 2020)

or the maser technique (Pesce et al. 2020), in both cases for which only a single galaxy

is available for comparison with Cepheids.

The immediate goal of this paper is to update the F20 TRGB calibration of H0,

which was based solely on a geometric distance to the Large Magellanic Cloud (LMC).

In the interim, a number of detailed new studies of the giant branch population in our

own and several nearby galaxies can now provide new and independent calibrations of

the TRGB. Five independent calibrations are examined in this paper. These include:

1. Observations of the TRGB in the outer halo of the maser galaxy, NGC 4258

(Jang et al. 2021).

2. Observations of TRGB stars in 46 Galactic globular clusters spanning a range

of metallicities (Cerny et al. 2020), calibrated via a Detached Eclipsing Binary

(DEB) distance to ω Cen.

3. A new geometric distance to the Small Magellanic Cloud (SMC) based on an

augmented sample of 15 DEBs (Graczyk et al. 2020), incorporating the up-

dated reddening and extinction maps of Skowron et al. (2021), together with

an updated measurement of the TRGB magnitude by tt (2021, in prep).

4. A re-analysis of the OGLE-III data for the LMC by Hoyt (2021), incorporating

the updated reddening and extinction maps of Skowron et al. (2021).

5. New Magellan imaging data for two Milky Way dwarf spheroidal galaxies, Sculp-

tor (Tran et al. 2021) and Fornax (Oakes et al. 2021), as well as HST/ACS

published data for four LMC globular clusters (Olsen et al. 1998) provide an

additional check on the calibration of the TRGB zero point.

A second goal of this paper is to examine and inter-compare recent calibrations

of the TRGB and Cepheid distance scales, and finally, a third goal is to assess the

significance of the tension in H0, as it currently stands.


The outline of this paper is as follows. In §2 we describe the recent calibrations

of the TRGB; in §3 we discuss the implications of these results in the context of the

determination of H0; in §4 we summarize recent calibrations of the Cepheid Leavitt

law. We then compare the H0 values in §5, and finally, in §6 we discuss the current

status, strengths and weaknesses in the TRGB and Cepheid distance scales, before

comparing our results with other methods in §7 and summarizing our results in §8.

In brief, based on four independent calibrations of the TRGB absolute magni-

tude, we find MTRGBF814W = −4.049 ± 0.015 (stat) ± 0.035 (sys) mag, leading to a value

of H0 = 69.8 ± 0.6 (stat) ± 1.6 (sys) km s−1 Mpc−1. Accurate calibration of the

extragalactic distance scale remains a challenging endeavor, and <1% measurements

of the CMB set a high (and currently not attainable) bar for the local distance scale

to match. The discrepancy in local (TRGB versus Cepheid) measurements suggests

that there are issues in the local distance scale that need to be understood before we

can unambiguously make extraordinary claims like new physics.

2. ABSOLUTE CALIBRATION OF THE TRGB

As can be seen from Table 1 the value of the absolute I-band magnitude of the

TRGB has remained quite stable over the 30 years in which it has been measured,

generally falling within the range of MI = -4.00 to -4.05 mag [at (V-I)o = 1.6 mag].

In this section, we present a summary of several independent calibrations of the

TRGB that have become available since the Freedman et al. (2020) calibration, which

was based solely on the DEB distance to the LMC. Importantly, these calibrations

are based on very different methods for measuring absolute distances, including a

geometric maser technique, geometric parallaxes and geometric DEB distances. In

§2.6, we combine all of these results to obtain an updated calibration of the TRGB.

These results are summarized in Table 3 in §2.6.

2.1. The Megamaser Galaxy NGC 4258

The nearby spiral galaxy NGC 4258, at a distance of 7.6 Mpc, is an excellent

target for providing a high-accuracy calibration of the TRGB. It is host to a sam-

ple of H2O megamasers, rotating within a highly-inclined (87◦) accretion disk about

a supermassive black hole, from which a geometric distance to the galaxy can be

measured (see Humphreys et al. 2013; Reid et al. 2019). The most recent geometric

distance to NGC 4258 is µ0 = 29.397 ± 0.024 (stat) ± 0.022 (sys) mag (Reid et al.

2019), a 1.5% measurement.

6 Freedman

Table 1. Absolute I-band TRGB calibrations

M ITRGB

a Reference

-4.0 ±0.1 Lee et al. (1993)

-4.05 b Rizzi et al. (2007)

-4.04 c Bellazzini (2008)

-4.05 ±0.02 ±0.10 Tammann et al. (2008)

-4.01 Bono et al. (2008b)

-4.03 d Madore et al. (2009)

-4.02 ±0.06 e Jang & Lee (2017)

-4.01 ±0.04 Reid et al. (2019)

-3.97 ± 0.046 Yuan et al. (2019)

-4.05 ±0.02 ±0.04 Freedman et al. (2020)

-4.04 ±0.01 ±0.03 f Hoyt (2021) : LMC

-4.05 ± 0.03 ±0.04 g Hoyt (2021) : SMC

a At (V-I) = 1.6 mag unless otherwise noted

b -4.05 + 0.217 × [(V-I) -1.6]

c -3.939 - 0.194 × (V-I) + 0.08 × (V-I)2

d -4.05 + 0.2 × [(V-I) - 1.5]

e -4.016 + 0.091 ×[(V-I)0 - 1.5]2 - 0.007 × [(V-I)0- 1.5]

f 1.60 < (V − I)0 < 1.95 mag

g 1.45 < (V − I)0 < 1.65 mag

The most extensive study of the TRGB in NGC 4258 has been published by

Jang et al. (2021). This measurement is based on a set of 15 archival HST/ACS

fields covering 54 square arcmin, located near the minor axis in the dust- and gas-free

outer halo of the galaxy. The analysis was further confined primarily to regions at

a de-projected semi-major axis distance of >14 arcmin (∼ 30 kpc) from the center

of the galaxy. The RGB stars at this large distance are well-separated from each

other, and are demonstrably free from crowding/blending effects. Moreover, these

halo RGB stars are relatively blue and metal poor, and do not exhibit a wide range

in color/metallicity. The wide areal coverage results in a well-populated giant branch

with about 3,000 red giant stars one-magnitude below the tip itself. As described

in detail in Jang et al., extensive tests for systematics were undertaken; for exam-

ple, using artificial stars; comparing DOLPHOT and DAOPHOT photometry; and

comparing results using different point spread functions, sky-fitting parameters, and

radial spatial cuts. Moreover, the HST/ACS data used for this study are on the

F814W flight magnitude system used in the F19 study, and thus do not require a

photometric transformation as for the case of the LMC zero point. Jang et al. obtain


a TRGB zero-point of MTRGB814 = -4.050 ± 0.028 ± 0.048, using the maser distance

determined by Reid et al. (2019). A detailed description of the error budget and the

adopted statistical and systematic uncertainties are given in §6 and Table 4 of Jang

et al. This independent TRGB calibration agrees to better than 1% with the value

of M814 = -4.054 mag found earlier by F20, as well as that of -4.045 mag measured

by Hoyt (2021), as described in §2.3.

Alternatively, if we instead determine the distance to NGC 4258 based on the

LMC TRGB calibration of Hoyt (2021), given the measured apparent TRGB mag-

nitude of mN4258814,o = 25.347 ± 0.014 ± 0.005 (Jang et al. 2021), we find a distance

modulus of µo = 29.392 ± 0.018 ± 0.032 mag. The agreement with the maser dis-

tance of 29.397 ± 0.033 mag (Reid et al. 2019) is at a level of better than 1%, differing

by <0.2σ. In contrast, we note that a Cepheid calibration of the distance to NGC

4258 does not as yield good agreement with that of the maser distance. As recently

described in Efstathiou (2020), a calibration of the Cepheid distance to NGC 4258

based on the LMC differs from the maser distance by 2.0-3.5σ, depending on the

adopted correction for metallicity. The Milky Way and NGC 4258 metallicities are

very similar, however, and should be independent of a metallicity effect. If instead,

the Milky Way is adopted as the anchor galaxy to determine the Cepheid distance

to NGC 4258, a distance modulus of 29.242 ± 0.052 is obtained, which differs from

the maser distance by 7% at a 2σ level of significance. We defer a discussion of the

implications of these differences to §5.

Finally, we note that the location of the fields studied by Jang et al. (2021) in

the outer halo of NGC 4258 is optimal for avoiding dust and gas, as well as being

separated from the high surface brightness galactic disk, thereby minimizing the level

of systematic effects that plague efforts to measure the TRGB in the star-forming

region of the disk of this galaxy, issues not considered, for example, in Macri et al.

(2006); Reid et al. (2019).

2.2. Galactic Globular Clusters

A second and completely independent method for calibrating the TRGB uses

photometry of well-measured giant branches in globular clusters within our own Milky

Way. Collectively, the Milky Way globular clusters span a wide range in metallicity,

which overlaps well with those measured for giant stars in the halos of nearby, resolved

galaxies.

This approach to calibrating the TRGB was first carried out by Da Costa & Ar-

mandroff (1990), using CCD imaging data for six globular clusters. That calibration,

for which distances were obtained using theoretical horizontal branch models from

Lee et al. (1990) (to calibrate the luminosities of RR Lyrae stars), formed the basis

8 Freedman

of the Lee et al. (1993) early application of the TRGB method to the extragalactic

distance scale. A decade later, Ferraro et al. (1999) assembled a homogenous sample

of 60 globular clusters, adopting the level of the theoretical zero-age horizontal branch

as the basis from which to measure absolute distances. As these authors noted, the

advantage of the horizontal branch is the simplicity of the measurement as compared

to RR Lyrae stars, for which variability and evolutionary effects need to be accounted

for, and for which uncertainties due to metallicity still remain. Bellazzini et al. (2001,

2004) based their calibration on observations of the two populous globular clusters,

ω Centauri and 47 Tucanae, calibrated using a DEB distance to ω Cen (Thompson

et al. 2001), and an average of literature distances for 47 Tuc. Subsequently, Rizzi

et al. (2007) based their distances on the well-developed horizontal branches of five

Local Group galaxies (IC 1613, NGC 185, Fornax, Sculptor and M33) spanning a

range in metallicities of -1.74 < [Fe/H] < -1.02 dex.

In a recent study, Cerny et al. (2020) have analyzed a sample of 46 low-reddening

[E(B-V) < 0.25 mag] Milky Way globular clusters with uniformly reduced photometry

available from Stetson et al. (2019) and through the Canadian Astronomy Data Center

(CADC).3 This 46-cluster catalog was then cross-matched to the Gaia Data Release

2 (DR2) database, and membership for these clusters was determined using the DR2

proper motion data and a Gaussian-mixture-model clustering algorithm. Preliminary

E(B − V ) reddening estimates and initial distance estimates were taken from Harris

(1996, 2010).

A composite MI versus (V − I)o color-magnitude diagram (CMD) is shown in

Figure 1 for the 46 low-reddening clusters from Cerny et al. (2020). This composite

shows a well-defined giant branch, sampling a wide range of metallicities from -2.4 <

[Fe/H] < -1.0 dex. As described in more detail in Cerny et al., high signal-to-noise and

low-extinction clusters were used to define a fiducial lower envelope to the blue and

red horizontal branches, and a maximum-likelihood grid search technique was used

to align the remaining clusters onto a common calibration. The zero point of the

calibration was set by the geometric DEB distance to ω Cen, measured by Thompson

et al. (2001). The resultant blue and red horizontal branches are shown in Figure 1.4

Applying a Sobel edge-detection filter to the composite luminosity function for the

TRGB, Cerny et al. determined an absolute I-band TRGB magnitude -4.056 mag,

which, following F19, transforms to flight magnitudes as M814W = -4.063 ± 0.07 ±0.11 mag.

3 The Stetson catalog is based on a collection of about 90,000 images for 48 clusters, all having UBV RIphotometry, for which a comparison of the different data sets constrains the photometric zero-pointuncertainties at the millimag level. Eleven of those clusters did not meet the Cerny et al. (2020)low-reddening criterion. Cerny et al. expanded the Stetson catalog to incorporate nine additionallow-reddening clusters with BV I photometry alone, archived at the CADC, and analyzed with thesame DAOPHOT/ALLFRAME software (Stetson 1987, 1994).

4 Note that the process of aligning the clusters based on their horizontal branches is completelyindependent of the TRGB.


Figure 1. A composite MI versus (V − I)o color-magnitude diagram based on 46 Galac-tic globular clusters, color-coded by the density of points. The clusters span a range inmetallicity of −2.4 < [Fe/H] < −1.0 dex. Cluster membership was determined from theirGaia DR2 proper motions. The red rectangular box outlines the region of the red giantbranch that is expanded in Figure 9. The horizontal branch, main-sequence turnoff andgiant branch are labeled. The horizontal gray dashed line indicates the TRGB at MI =-4.056 mag, and the cyan and red lines indicate the blue and red horizontal branch fits asmeasured by Cerny et al. (2020).

10 Freedman

2.2.1. Gaia Early Data Release 3 Calibration of Galactic Globular Clusters

With the ESA Gaia mission, the promise of astrometry reaching tens of mi-

croarcsecond accuracy (Gaia Collaboration, Prusti et al. 2016) has been eagerly

anticipated. Such astrometry for Galactic Cepheids, TRGB stars and other distance

indicators will ultimately fix the absolute zero point of the extragalactic distance scale

to an unprecedented accuracy of better than 1%. However, in early data releases, it

was discovered that there is a zero-point offset (e.g., Lindegren et al. 2016). This

offset results from the fact that the basic angle between the two Gaia telescopes is

varying (resulting in a degeneracy with the absolute parallax). In addition, these

variations lead to zero-point corrections that are a function of the magnitude, color,

and position of the star on the sky (Lindegren et al. 2018; Arenou et al. 2018). In

DR2, Mignard et al. (2018) and Arenou et al. (2018) found an average zero-point

offset of -29 µas relative to the background reference frame for more than 550,000

quasars defined by the International Celestial Reference System.

Recently, the Gaia mission has released a new and updated database (Early

Data Release 3; EDR3). This Gaia EDR3 database (Gaia Collaboration et al. 2021)

contains parallaxes, proper motions, positions and photometry for 1.8 billion sources

brighter than magnitude G=21 mag (Lindegren et al. 2021b). The baseline for EDR3

is 34 months compared to 22 months for DR2, and thus provides a significant improve-

ment to the astrometry. The parallax improvement is estimated to be 20% compared

to DR2; in addition, the variance in the parallaxes (the systematic uncertainty), as

measured over the sky and estimated from quasars, has been reduced by 30–40%

(Gaia Collaboration et al. 2021). Still, on average, the zero-point offset for EDR3 is

found to be -17 µas (in the sense that the Gaia parallaxes are too small). The Gaia

collaboration has provided additional parallax corrections for EDR3, which are again

a function of G magnitude, color and ecliptic latitude (Lindegren et al. 2021a).

However, as the Gaia Collaboration emphasizes (e.g., Bailer-Jones et al. 2021;

Fabricius et al. 2021) there is a significant variance in these measured offsets over the

sky, and the EDR3 uncertainties in the parallaxes for different objects are correlated

as a function of their angular separations. Lindegren et al. (2021a,b) calculate the

angular power spectrum of parallax systematic biases in Gaia EDR3 quasar data

and estimate that the rms variation of the parallax systematics (excluding the global

offset) is about 10 µas on angular scales >∼10 degrees. More recently, Maız Apellaniz

et al. (2021) and Vasiliev & Baumgardt (2021) have analyzed EDR3 parallax data

for a sample of Milky Way globular clusters. Both studies concur with the result

that there are significant rms variations on both large and small angular scales. Maız

Apellaniz et al. conclude that the angular covariance limit results in a minimum (and

systematic) uncertainty for EDR3 parallaxes for individual stars or small-angular

diameter clusters of 10.3 µas out to 30 arcmin. The rms fluctuations can reach as


high as 30-50 µas. They further note that the uncertainty cannot be significantly

reduced for larger clusters.

The minimum 10 µas systematic uncertainty in the EDR3 parallaxes limits the

accuracy with which we can calibrate the TRGB for Galactic globular clusters. Cerny

et al. (2020) (as described in §2.2 above) based their calibration on the geometric DEB

distance to ω Cen, anticipating that in future, accurate Gaia parallax measurements

for all 46 clusters will be available for calibration. ω Cen has a measured Gaia

EDR3 parallax of 189 µas or a distance of 5.25+0.28−0.25 kpc (Maız Apellaniz et al. 2021;

Vasiliev & Baumgardt 2021). Unfortunately, a minimum systematic uncertainty of

10 µas results in a minimum (large) distance uncertainty of 5% (0.1 mag) for ω

Cen. Additionally concerning, Vasiliev & Baumgardt provide evidence that the Gaia

distances are systematically (and significantly) smaller than the previously published

distances to these systems (the parallaxes are overestimated by 6-9 µas above the

correction provided by Lindegren et al. (2021a)).

Based on Gaia EDR3 measurements for ω Cen, Soltis et al. (2021) more opti-

mistically quote a parallax measurement of 0.191 ± 0.001 (statistical) ± 0.004 (sys-

tematic) mas (2.2% total uncertainty) corresponding to a distance of 5.24 ± 0.11

kpc, an uncertainty significantly smaller than (the minimum of 5%) demonstrated by

all of the studies discussed above. As Vasiliev & Baumgardt (2021) note, these rms

variations across the sky are irreducible at present and they thus conclude that the

true uncertainty of the Soltis et al. result has been significantly underestimated.

Further independent constraints on the distance to ω Cen come from measure-

ments of the RR Lyrae stars in the cluster. Recent near-infrared JHK measurements

by Braga et al. (2018) result in distances of 5.43-5.49 kpc (depending on their metal-

licity calibration) with quoted total uncertainties of 2%, in good agreement with the

DEB distance, as well as with a number of other published optical and near-infrared

RR Lyrae measurements listed in their Table 8. To within the 1-σ uncertainties, the

recent RR Lyrae distance scale agrees with the Gaia EDR3 measurements of Maız

Apellaniz et al. (2021) and Vasiliev & Baumgardt (2021).5

The uncertainties (of order 5%) in both the DEB and Gaia EDR3 distances for ω

Cen are currently too large to provide the 1% level of accuracy that will ultimately be

required for a resolution of the tension in H0. For this paper, we adopt the Cerny et al.

(2020) calibration, with a distance of 5.44 kpc, and its (large) associated uncertainty

of ±5% (±0.1 mag). As a result, it receives a lower weight in the determination of

the value of H0 described in §3. We note that adopting the Gaia EDR3 distance of

5 More recently, Baumgardt & Vasiliev (2021) obtain a 1% distance to ω Cen by combining CMDfitting, RR Lyrae, DEBs, in addition to the new Gaia EDR3 distance, corrected for the systematicoffset. They find a distance of 5.426 ± 0.047 kpc (their Table 2), in excellent agreement with theresults presented here.

12 Freedman

5.25 kpc with the same uncertainty of ±5% increases H0 by only 0.1% in the final

analysis.

The Gaia parallaxes and additional measurements will continue to improve as

longer time baselines are established over the course of the mission: the full potential

of Gaia has yet to be realized. DR4 and DR5 are expected to be based on 5.5 and

10 years of data, respectively6.

2.3. LMC Calibration

F20 measured the TRGB for the LMC using the OGLE “Shallow” survey data

of Ulaczyk et al. (2012)7. In order to avoid crowding/blending effects within the high-

surface-brightness bar, the sample of stars analyzed was confined to stars outside of

a circle of one degree radius, centered on the bar of the LMC. The LMC reddening

and extinction were measured using VIJHK photometry, differentially with respect

to two low-reddening galaxies, IC 1613 and the SMC. Based on the DEB (Pietrzynski

2019) distance modulus to the LMC of 18.477 mag, the extinction-corrected absolute

magnitude of the TRGB for the I band was found to be MTRGBI = -4.047± 0.022 (stat)

± 0.039 (sys) mag. The Pietrzynski measurement is based on the surface-brightness-

color calibration for late-type giant stars, from which the angular diameters of giant

stars can be measured to an accuracy of 0.8%.

Recently, Hoyt (2021) has undertaken a detailed remeasurement of the LMC

TRGB based on OGLE-III photometry, isolating regions where the edge-detection

measurements are sharp and single-peaked. He illustrates that these same regions are

also low in dust content, and located away from regions of star formation. He incor-

porates the new reddening and extinction maps of Skowron et al. (2021) determined

from the colors of red clump stars based on OGLE-IV photometry. Adopting the 1%

distance to the LMC based on DEBs (Pietrzynski 2019), he finds MTRGBI = -4.038 ±

0.012 (stat) ± 0.032 (sys) mag, consistent to within 1% with the earlier results. A

detailed description of the error budget and the adopted statistical and systematic

uncertainties is given in his Table 3. The systematic uncertainty includes a ±0.01

mag term on the OGLE photometric zero point. An additional ±0.01 mag system-

atic uncertainty is included in the ground-to-HST calibration resulting in MTRGB814 =

-4.045± 0.012± 0.034 mag.

As an aside, we note that Yuan et al. (2019) argued that the F19 calibration of

H0 based on the distance to the LMC was in error. However, Freedman et al. (2020)

and Hoyt (2021) describe in some detail a number of incorrect assumptions that were

made by Yuan et al. The excellent agreement found here between the completely

6 https://www.cosmos.esa.int/web/gaia/science-performance7 The LMC data are available at http://www.astrouw.edu.pl/ogle/ ogle3/maps/


independent LMC, NGC 4258, SMC and Galactic globular-cluster calibrations argues

even more strongly against the claims made in Yuan et al. Moreover, even if the LMC

were to be excluded from the TRGB calibration altogether, the resulting change in

the overall value of H0 is insignificant (<1%).

2.4. The Small Magellanic Cloud (SMC)

The interaction of the LMC and SMC has resulted in a tidally-extended structure

to the SMC, which has historically complicated the measurement of the SMC distance.

F20 measured the TRGB using published OGLE data8 for the inner region of the

SMC, thereby avoiding confusion with the more extended tidal tails. They measured

an I-band magnitude for the TRGB of mTRGBI = 14.93 mag, adopting a foreground

extinction value of AI = 0.056 mag.9

Mapping out the inclined system with very high precision, Graczyk et al. (2020)

have recently measured a new DEB distance to the central region of the SMC to

an accuracy of better than 2%, based on the surface-brightness-color calibration of

(Pietrzynski 2019). Augmenting the sample of measured DEBs from their previously

published sample (from 5 to 15, a three-fold increase), Graczyk et al. (2020) determine

a distance modulus of µ0 = 18.977 ± 0.016 (stat) ± 0.028 (sys) mag. The SMC

thus provides another opportunity for an updated and independent calibration of the

TRGB. An advantage of the SMC is its low star formation rate and dust content.

Hoyt (2021) has also undertaken an reanalysis of the SMC OGLE-III data in-

corporating the updated Skowron et al. (2021) reddening maps. He measures an ap-

parent tip magnitude of mTRGBI = 14.93 mag. A detailed description of the adopted

statistical and systematic uncertainties is given in his Table 3. Based on the new

Graczyk et al. (2020) true DEB distance modulus he finds MTRGBI = -4.050 ± 0.030

(stat) ± 0.040 (sys) mag, in excellent agreement with the NGC 4258, Milky Way

globular-cluster, and the LMC calibrations discussed above. An additional ±0.01

mag systematic uncertainty is included in the ground-to-HST calibration resulting in

MTRGB814 = -4.057± 0.030± 0.040 mag.

2.5. Additional Comparisons

In the cases described in this section, we do not use these systems to calibrate

H0, but rather note their excellent consistency with the other calibrations presented

here, lending further confidence to the overall calibration of the TRGB.

8 The SMC OGLE data are available at website http://www.astrouw.edu.pl/ogle/ogle3/maps/.9 The value quoted in Freedman et al. (2020) is for the extinction-corrected ITRGB

o and not for the ap-parent magnitude as stated. Adopting the distance modulus based on five previously-measured DEBmeasurements (which yielded a value of µ0 = 18.965 mag), would result in a zero-point calibrationfor the TRGB of MTRGB

I = -4.035 ± 0.03 (stat) ± 0.05 (sys) mag.

14 Freedman

Table 2. Data for LMC clusters

Cluster µo E(V-I) AI

NGC 2005 18.58 0.139 0.170

NGC 2019 18.57 0.083 0.102

NGC 1754 18.87 0.125 0.153

NGC 1835 18.48 0.111 0.136

Two recent studies of the Sculptor (Tran et al. 2021, in prep.) and Fornax

(Oakes et al. 2021, in prep.) dwarf spheroidal companions to the Milky Way provide

additional calibrations of the TRGB, constituting consistency checks on the geomet-

ric calibrations (for the LMC, Milky Way, NGC 4258 and the SMC) described above.

Wide-field Magellan IMACS VI data were obtained for each galaxy, from which the

position of the apparent TRGB, and the position of the horizontal branch were mea-

sured. Tran et al. measured an extinction-corrected value of the apparent TRGB

I-band magnitude for Sculptor of mTRGBIo

= 15.487 ± 0.057 ± 0.014 mag. For For-

nax, Oakes et al. found mTRGBIo

= 16.75 ± 0.03 ± 0.01 mag. Adopting the absolute

calibration of the horizontal branch from Cerny et al. (2020), as described in §2.2

above, and shown plotted in Figure 1, yields true distance moduli of 19.56 ± 0.03

± 0.10 mag and 20.79 ± 0.02 ± 0.10 mag, for Sculptor and Fornax, respectively.

These measurements yield absolute I-band calibrations of the TRGB (based on the

horizontal branch) of -4.07 ± 0.06 ± 0.10 and -4.04 ± 0.04 ± 0.10 mag, again in

excellent agreement with the independent calibrations based on NGC 4258, the LMC

and the SMC.

Finally, we have also examined the F814W and F555W HST/ACS data obtained

by Olsen et al. (1998) for a number of globular clusters in the LMC: specifically NGC

1754, NGC 1835, NGC 2005, NGC 2019. Table 2 lists the reddenings and extinc-

tions measured for each cluster by Olsen et al., and the true distance moduli based

on the horizontal branch calibration of Cerny et al. (2020). We show a composite

color-magnitude diagram for these objects in Figure 2. Adopting the Cerny et al.

calibration results in a measured TRGB magnitude of -4.085 ± 0.05 ± 0.10 mag.

As noted previously, these systems are not of comparable accuracy (or indepen-

dence) to yield an independent calibration of H0, but their consistency, to within the

uncertainties, already provides a further test of the robustness of the TRGB calibra-

tion. In future, when parallaxes accurate to 1% become available for a large sample

of Milky Way globular clusters, these horizontal-branch measurements will become a

powerful independent route to a calibration of the TRGB.


−0.5 0.0 0.5 1.0 1.5 2.0

[F555W]-[F814W]

−5

−4

−3

−2

−1

0

1

2

[F814W

]

TRGB

Composite LMC Globular Cluster Giant Branch CMD

NGC 1835

NGC 2005

NGC 2019

NGC 1754

Figure 2. I versus (V −I) color-magnitude diagrams for four LMC globular clusters basedon HST/ACS data from Olsen et al. (1998). The blue and red fiducial horizontal branchesdefined by Cerny et al. (2020) are shown. The position of the tip and 1-σ uncertainties areillustrated by the solid and dashed horizontal lines at the top of the figure.

2.6. Adopted TRGB Calibration

Table 3 lists the TRGB absolute magnitude at F814W for the geometric calibra-

tions described above. Where the calibration was carried out for ground-based data

(as for the LMC, SMC and the Milky Way clusters), these have been transformed

to the HST/ACS F814W flight-magnitude system. The NGC 4258 calibration was

carried out entirely with HST and is already on the F814W flight magnitude system.

As discussed in F19 and F20, the transformation from the I-band to F814W results

in a zero point that is brighter by -0.0068 mag. As can be seen from this table, the

good agreement of the TRGB zero point based on the calibrations for many anchors

means that the adoption or rejection of a particular galaxy does not significantly

impact the overall result.

Figure 3 shows the relative probability density functions (PDFs) for the absolute

TRGB F814W magnitudes discussed in §2 above. Here we separate the contributions

of the statistical and systematic errors in each case, so that the contribution of both

types of uncertainties can be clearly seen. In Figure 3a), the widths of the Gaus-

sians represent the individual statistical errors in each determination only, whereas

16 Freedman

Table 3. TRGB zero-point calibration

Object MTRGBF814W (mag) σstat σsys Reference

NGC 4258 a -4.050 0.028 0.048 Jang et al. (2021)

Milky Way globular clusters b -4.063c 0.07 0.11 Cerny et al. (2020)

LMC d -4.045c 0.012 0.034 Hoyt (2021)

SMC e -4.057c 0.030 0.040 Hoyt (2021)

Sculptor f -4.08c 0.06 0.11 Tran et al. (2021)

Fornax g -4.05c 0.04 0.11 Oakes et al. (2021)

LMC globular clusters h -4.085 0.05 0.10 this paper

Adopted Value (MW, NGC 4258, LMC, SMC) -4.049 0.015 0.035 this paper (§2.6)

a H2O Megamaser distance calibrationb Optical data, Gaia proper motion selection; ω Cen DEB calibration; MI = -4.056 mag.c Transformation to MTRGB

F814W = MI - 0.0068 mag following Freedman et al. (2019).d LMC DEB calibration; MI = -4.038 mag. An additional ± systematic uncertainty is included in

the ground-to-HST calibration.e SMC DEB calibration ; MI = -4.050 mag. An additional ± systematic uncertainty is included in

the ground-to-HST calibration.f ω Cen DEB calibration ; MI = -4.07 magg ω Cen DEB calibration ; MI = -4.04 magh ω Cen DEB calibration

the systematic uncertainties are illustrated separately by the error bars at the top

of the plot (using the same color coding) for each object. Conversely, in Figure 3b),

the widths of the PDFs represent the individual systematic errors in each determina-

tion only, whereas the statistical uncertainties are illustrated separately by the error

bars at the top of the plot. The integrals of the PDFs for the LMC, Milky Way,

NGC 4258 and the SMC each have unit area. The statistical and systematic errors

for each individual determination, σi, are given by the 16th and 84th percentiles of the

Gaussians in Figures 3a and b, respectively. The Frequentist sums of the probability

distributions are shown in both cases by the black lines. For the total sample, σmean

=∑σi/√

(N − 1), where N = 4.


As we have seen, the Milky Way TRGB magnitude is based on a sample of

46 clusters calibrated to the DEB distance to ω Cen. The calibrations of Sculptor,

Fornax and the LMC clusters are not independent, however, since they all rely on

the Milky Way calibration of the horizontal branch. Moreover, Sculptor and Fornax

are single objects, and the TRGB for the LMC clusters is sparsely populated. For

illustrative purposes, in Figures 3a) and b), the areas for these Gaussians have thus

been down-weighted by a factor 1/f as shown in Equation 1:

1

f√

2πσ2e−0.5

(x−<x>σ2

)(1)

where f =√

(N), and N = 46, the size of the Milky Way globular cluster sample.

Thus Sculptor, Fornax and the LMC clusters do not contribute to the adopted overall

calibration, but they do provide a consistency check on the horizontal branch to

TRGB distance scale.

The Frequentist sums of the probability distributions are shown by the black

lines in Figure 4a. The mode of the summed distribution for the four primary TRGB

calibrators is -4.049 mag. As shown in Figure 4b, an identical result is obtained for

a Bayesian analysis (albeit with smaller uncertainty), in which a uniform prior is

adopted, and the product of the distributions is determined. In addition, a simple

weighted average for the LMC, Milky Way, NGC 4258 and the SMC also gives a

result to within 0.001 mag of -4.049 mag. We adopt this robust value, MTRGBF814W =

−4.049 ± 0.015 (stat) ± 0.035 (sys) mag, for the absolute magnitude of the TRGB.

The (exact) agreement of the various means of combining the four calibrations lends

confidence to the overall result; i.e., it is independent of the choice of statistical

approach adopted to combine the results. Finally, we note that this value agrees to

better than 1% with that given by F20, who found MTRGBF814W = -4.054 ± 0.022 ± 0.039

mag.

3. THE HUBBLE CONSTANT BASED ON THE TRGB

3.1. New TRGB Calibration of Ho Based on Supernovae Ia

We turn now to a determination of H0 based on the TRGB calibration discussed

in §2.6. This is an update of the calibration of H0 presented in F19 and F20, applied to

the Carnegie Supernova Project (CSP) sample of 99 SNe Ia observed at high cadence

and multiple wavelengths (Krisciunas et al. 2017). That measurement of H0 was based

on HST/ACS observations of the halos of 15 galaxies that were hosts to 18 SNe Ia.

The measured absolute I-band magnitude of the TRGB from Freedman et al. (2020)

was MF814W = -4.054 ± 0.022 (stat) ± 0.039 (sys) mag, tied to the geometric DEB

18 Freedman

−4.3−4.2−4.1−4.0−3.9−3.8M814W

Rela

tive

Pro

bab

ilit

y

M814W for TRGB AnchorsSystematic Error Bars and Statistical Distributions

LMC

Milky Way

NGC 4258

SMC

Sculptor

Fornax

LMC clusters

Re-scaled sum

Systematic Error Bars

Statistical ErrorDistributions

M814W = -4.048+0.010 mag−0.020 mag[σmean (stat)]

a)

−4.3−4.2−4.1−4.0−3.9−3.8M814W

Rela

tive

Pro

bab

ilit

yD

en

sity

M814W for TRGB AnchorsStatistical Error Bars and Systematic Distributions

LMC

Milky Way

NGC 4258

SMC

Sculptor

Fornax

LMC clusters

Re-scaled sum

Statistical Error Barsb)

Systematic ErrorDistributions

M814W = -4.050±0.034 mag[σmean (sys)]

Figure 3. Probability density functions for the measured absolute magnitude of the TRGB.The statistical and systematic errors are shown separately, so that the relative contributionsof each can be easily seen for each galaxy. The statistical uncertainties can be improved byincreasing the sample size in future, decreasing as 1 /

√(N). In Figure 3a), the systematic

error bars are shown at the top of the plot, and the statistical error distributions are shownat the bottom. In Figure 3b), the statistical error bars are shown at the top, and thesystematic error distributions are shown at the bottom. As discussed in §6.3, there is somecovariance in the systematic uncertainties. Shown are the sum of all of the PDFs (black),the LMC (red), Milky Way (blue), NGC 4258 (purple), SMC (orange), Sculptor (magenta),Fornax (green) and the composite of the four LMC globular clusters (cyan). The resultsfor Sculptor, Fornax and the LMC clusters are shown for comparison purposes only. Thestatistical and systematic errors on the mean are labeled, along with the adopted value ofMTRGB

F814W = −4.049 ± 0.015 (stat) ± 0.035 (sys) mag, consistent, to within 0.001 mag, withthe mode of the summed distribution in each case.


−4.3−4.2−4.1−4.0−3.9−3.8M814W

Rela

tive

Pro

bab

ilit

yD

en

sity

Total (Frequentist) ErrorsAbsolute TRGB Calibration M814W

LMC

Milky Way

NGC 4258

SMC

Sculptor

Fornax

LMC clusters

Re-scaled sum

a)

M814W = -4.049

± 0.015 (stat)± 0.035 (sys)

−4.3−4.2−4.1−4.0−3.9−3.8M814W

Rela

tive

Pro

bab

ilit

yD

en

sity

Total (Bayesian) ErrorsAbsolute TRGB Calibration M814W

LMC

MW clusters

NGC 4258

SMC

Sculptor

Fornax

LMC clusters

Product of PDFs

b)

M814W = -4.049

± 0.010 (stat)± 0.022 (sys)

Figure 4. Probability density functions for the measured absolute magnitude of the TRGB.The total errors (statistical and systematic, combined in quadrature) are shown, as describedin the text. Figure 4a) The Frequentist sum of all of the PDFs (black), the LMC (red),Milky Way (blue), NGC 4258 (purple), SMC (orange), Sculptor (magenta), Fornax (green)and the composite of the four LMC globular clusters (cyan). The statistical and systematicerrors on the mean are labeled, as described in the text, along with the adopted value ofMTRGB

F814W = −4.049± 0.015 (stat)± 0.035 (sys) mag. Figure 4b) The product of the PDFs.Color scheme is the same as that for Figure 3.

distance modulus to the LMC from Pietrzynski (2019) of 18.477 mag. We now use

four independent calibrations (NGC 4258, Milky Way, LMC and SMC) superseding

the single calibration based on the LMC alone.

To briefly summarize, in F19 the CSP analysis was undertaken with the SNooPy

package (Burns et al. 2018), which characterizes the SNe Ia light-curve shape us-

ing a color-stretch parameter, sBV . Magnitudes were computed using two different

20 Freedman

approaches to the reddening where

B′ = B − P 1(sBV − 1)− P 2(sBV − 1)2 − CT − αM(log10M∗/M� −M0), (2)

where P 1 is the linear coefficient and P 2 is the quadratic coefficient in (sBV −1); B and

V are the apparent, K-corrected peak magnitudes; αM is the slope of the correlation

between peak luminosity and host stellar mass M∗; and CT denotes the color term

for the two approaches. In the first case, CT = β(B-V), where a color coefficient

β results in a reddening-free magnitude, an approach originally proposed by Tripp

(1998). In the second approach, CT = RB E(B-V), where RB, the ratio of total-to-

selective absorption, and the reddening, E(B-V), are solved for explicitly using both

optical and near-infrared colors for each SN Ia. Using the MCMC fitter described

in Burns et al. (2018) and F19, which uses the “No U-Turn Sampler” from the data

modeling language STAN (Carpenter et al. 2017), and solving for the parameters in

Equation 2, the value of H0 and its error were obtained using both approaches.

As described in Hoyt et al. (2021), two new galaxies with directly measured

TRGB distances have been added to the CCHP sample since F19 and F20. NGC 5643

is host to SN 2013aa and SN 2017cbv (Burns et al. 2020); and NGC 1404, a member

of the Fornax cluster, is host to SN 2007on and SN 2011iv (Gall et al. 2018). Hoyt

et al. (2021) find that SN 2007on appears to be significantly underluminous, and it is

therefore excluded from the current analysis. In F19, the distance to NGC 1404 was

taken to be the average value given by the two other Fornax galaxies in the CCHP

sample, NGC 1316 and NGC 1365. In this paper, we adopt the new direct distance

to NGC 1404 (Hoyt et al. 2021) for SN 2011iv and add the two additional SNe Ia

in NGC 5643, augmenting the sample of 18 SNe Ia described in F19 to an updated

sample of 19. All distances are calibrated adopting MTRGBF814W = −4.049±0.015 (stat)±

0.035 (sys) mag, and used as new input to the MCMC analysis described in F19 (C.

Burns, priv. comm.).

In Table 4, we give the values of H0 and uncertainties obtained adopting the

new calibration of MTRGBF814W . Listed are H0 values based on both the CSP B-band and

H-band SNe Ia magnitudes for different color and dust reddening constraints. The

uncertainties (for the SNe Ia analysis alone) are determined from a diagonal covari-

ance matrix with respect to the TRGB distances. Following F19, we present results

applying both the Tripp and explicit E(B-V) reddening corrections. In addition, we

present the results adopting host-galaxy masses from Burns et al. (2018) originally

used in F19, as well as those measured in a more recent study of Uddin et al. (2020).

Figure 5 shows these results in flowchart form.

The values presented in Table 4 and Figure 5 represent different choices for: the

SN Ia sample (color and stretch); dealing with dust (Tripp versus E(B-V)); bandpass

for the SN Ia magnitudes (B versus H); and host galaxy-mass peak SN Ia luminosity


Table 4. Values of H0 (km s−1Mpc−1) for various choices of fit

Tripp E(B − V )

Band H0 (CSP18)a σ H0 (CSP20)b σ H0 (CSP18)a σ H0 (CSP20)b σ

Full Sample

B 69.48 1.39 69.88 1.25 70.75 1.32 71.50 1.29

H 69.13 1.35 70.48 1.23 69.36 1.46 70.33 1.41

sBV > 0.5 and (B − V ) < 0.5

B 69.38 1.36 69.57 1.24 69.39 1.04 70.04 1.05

H 68.80 1.34 70.00 1.25 68.47 1.44 69.47 1.42a CSP SNe Ia host-galaxy-mass corrections from Burns et al. (2018).b CSP SNe Ia host-galaxy-mass corrections from Uddin et al. (2020).

LMC Milky Way NGC 4258 SMC

TRGB Zero Point

-4.045± 0.012 ±0.034

-4.063± 0.07 ±0.11

-4.057± 0.030 ±0.040

-4.050± 0.028±0.048

Type Ia SupernovaeHo Excluding Red Fast Decliners

(sBV <0.5, E(B-V) < 0.5, (B-V) < 0.5)Ho Including Red Fast Decliners

(Full Sample)

Tripp. E(B-V)

H

69.38±1.36 69.39±1.0469.57±1.24 70.04±1.05

BurnsUddin

68.80±1.34 68.47± 1.44 70.00±1.25 69.47± 1.42

BurnsUddin

69.48±1.39 70.75±1.32 69.88±1.25 71.50±1.29

69.13±1.35 69.36±1.4669.88±1.25 70.33±1.41

Adopted Ho = 69.8 ±". $ (stat) ±%. $ (sys) km s-1 Mpc-1

Hubble Constant

M814W = -4.049 ± 0.015 ± 0.035 mag

Tripp. E(B-V)

B B

H

Figure 5. An overall flowchart summarizing the results of the TRGB zero-point calibrationdescribed in §2 and the SNe Ia calibration described in §3, leading to the adopted value ofH0 = 69.8 ± 0.6 (stat) ± 1.6 (sys) km s−1 Mpc−1. The TRGB zero-point is based on theM814W calibrations for the LMC, SMC, NGC 4258 and the Milky Way. The adopted valueof H0 is based on a sample of SNe Ia restricted to those with sBV > 0.5 and (B - V) <0.5, for which there is good proportional overlap between the TRGB and more distant hostgalaxy samples.

correlation (Burns versus Uddin). The various choices result in a full range in H0

values from 68.47 to 71.50 km s−1 Mpc−1. In selecting a best value of H0 from those

listed in Table 4, we select (following F19) the sample that minimizes the difference

22 Freedman

0.00 0.02 0.04 0.06 0.08zcmb

0

5

10

15

20

0.2 0.4 0.6 0.8 1.0 1.2sBV

0

5

10

15

20

0.0 0.5 1.0 1.5B V (mag)

0

10

20

30

40sBV < 0.5, B V > 0.5slowblueTRGB

8 9 10 11log(M/Msun)

0

10

20

30

Figure 6. The upper left panel shows the redshift distribution for the total sample ofCSP SNe Ia. In blue are the slow decliners (with sBV > 0.5 and (B - V) < 0.5), labeled“slowblue”. In orange are the red, fast decliners, and the nearby calibrating galaxies withmeasured TRGB distances are shown in green. The upper right panel shows the distributionof stretch values, and the lower two panels show the distributions of (B-V) and log10(

MM�

),

respectively. The “Full Sample” in Table 4 includes both the orange and blue distributions(i.e., the different samples are not overplotted, and no orange bins are being lost). Thegreen TRGB distribution is well-matched to that of the of slower, bluer decliners, anddoes not exhibit the extended tails seen in orange for stretch (with sBV < 0.5) and color((B − V ) > 0.5) of the red, fast decliners.

between the calibrator sample and the distant sample in terms of the nuisance vari-

ables: color, stretch, and host mass. In the histograms in Figure 6 we illustrate the

characteristics of the SN Ia in the distant galaxy sample compared with those for the

TRGB calibrators. The TRGB sample is shown in green; the overall CSP sample is

divided such that the blue, slow decliners (with sBV > 0.5 and (B - V) < 0.5) are

shown in blue (and labeled “slowblue”) and those with fast decline rates and redder

colors are shown in orange. In terms of stretch and color, the tails seen in orange (ex-

tending to sBV < 0.5 and (B−V ) > 0.5) are absent in the TRGB calibrating sample.

Unambiguously, in terms of stretch and color, the sample with the best overlap of

calibrator and distant SNe Ia is that of “slowblue”. This sample also overlaps well in

terms of host-galaxy mass, an advantage of the TRGB method, which can be applied

to both early- and late-type galaxies. (Cepheid variables are young objects found

only in star-forming (e.g., spiral) galaxies and cannot calibrate the SNe Ia found in

elliptical or S0 galaxies.)

We note the following:


1. Using B-band photometry, restricting the sample to that with sBV > 0.5 and (B

- V) < 0.5 (“slowblue”), and basing the analysis on the more recent Uddin et al.

(2020) host-galaxy masses results in a value of H0 = 69.57 ± 1.24 km s−1 Mpc−1

using the Tripp method and H0 = 70.04 ± 1.05 km s−1 Mpc−1 explicitly cor-

recting for dust (E(B-V)). Using H-band photometry, respectively results in

similar values of H0 = 70.00 ± 1.25 and H0 = 69.47 ± 1.42 km s−1 Mpc−1.

The corresponding values based on the Burns et al. (2018) masses are slightly

lower. The difference arises primarily because the slope of the mass correlation

in the optical is steeper for the Burns masses, whereas for the Uddin masses,

the relation is nearly flat for all filters.

2. The H-band data have the advantage of smaller dependence on the reddening, as

the correction (Rλ) is smaller, but they have the disadvantage of larger variance

because the sample of SNe Ia having H-band photometry is smaller. (The “Full

Sample” has 147 objects; “slowblue” restricts the sample to 129 objects; and

restricting the sample to those with H-band photometry results in 102 objects.)

3. The largest value of H0 (71.5) is obtained when the redder, faster decliners are

included in the analysis (the “Full Sample”). However, as noted above, these

solutions are strongly disfavored since there are no redder, faster decliners in

the more distant sample. In a broader context, no solution here reaches a value

as high as 74 km s−1 Mpc−1.

Although the differences in these H0 values are small (a total range of only 3

km s−1 Mpc−1), they illustrate the effect of different choices in the host-galaxy mass

correlation and method/filters adopted to correct for dust.

Our adopted best-fit value is based on 1) the sample of SNe Ia for which the

nuisance variables (color, stretch, and host mass) are comparable for the calibrating

TRGB galaxies and the distant SNe Ia; 2) an average of the Tripp/E(B-V) deter-

minations; 3) the recent host-galaxy masses measured by Uddin et al. (2020). We

choose the B-band measurements because the sample of SNe Ia is largest, and the

scatter for the H-band measurements is 40% larger (or a factor of two in the variance,

in the case of the E(B-V) correction). We adopt a best-fit value of 69.8 ± 1.2 (sys)

km s−1 Mpc−1. This latter uncertainty takes into account the systematic uncertainties

in the SN Ia analysis alone, without yet combining it with the TRGB zero-point sys-

tematic error. As discussed in Burns et al. (2018); Freedman et al. (2019), all of the

correction factors to the SN Ia light curves (P 1, P 2, sBV − 1, αM , β, E(B− V ), RB),

as described in §2 are computed; these then provide corrected magnitudes and a full

covariance matrix, used to determine H0 and the uncertainty given in Table 4. The

total uncertainty adopted for H0, including the uncertainty in the TRGB calibration

is discussed below.

24 Freedman

Table 5. H0 Values for Common TRGB and Cepheid Calibrators

Calibrator H0 (TRGB) H0 (Cepheids)a Cepheid Reference

LMC 69.9 ± 0.5 (stat) ± 1.6 (sys) 74.22 ± 1.82 Riess et al. (2019)

NGC 4258 69.7 ± 1.0 (stat) ± 2.0 (sys) 72.0 ± 1.9 Reid et al. (2019)

Milky Way 69.3 ± 0.8 (stat) ± 3.5 (sys) 73.0 ± 1.4 Riess et al. (2021)

SMC 69.5 ± 1.0 (stat) ± 1.7 (sys) ... ...

Adopted Value 69.8 ± 0.6 (stat) ± 1.6 (sys) 73.2 ± 1.3 Riess et al. (2021)

a The published SHoES H0 results are given with total errors only.

Table 6. Summary of H0 Uncertainties

Source of Error Random Error Systematic Error Description

TRGB Zero Point 0.7% 1.6% §2.6

CSP-I SNe Ia 0.5% 1.7% F19, §3

Total 0.9% 2.3% In quadrature

The H0 values and uncertainties based individually on the new TRGB calibra-

tions for NGC 4258 (§2.1), Galactic globular clusters (§2.2), the SMC (§2.4), and

the LMC §2.3 are listed in Table 5. For comparison, also listed are the H0 values,

their uncertainties and their published references from the SHoES team, based on the

Cepheid calibrations for the LMC, NGC 4258 and the Milky Way.

Both statistical and systematic uncertainties are given for the TRGB H0 deter-

minations in Table 5. The error bars include both the uncertainties for the TRGB

calibration discussed in §2.6 above, in addition to those arising from the calibra-

tion of the SNe Ia, as discussed above, and in F19. For the SNe Ia, the statisti-

cal uncertainty amounts to ± 0.5% with a systematic uncertainty of ±1.7%. The

final percentage errors are summarized in Table 6, with a final adopted value of

H0 = 69.8± 0.6 (stat)± 1.6 (sys) km s−1 Mpc−1.

Figure 7 shows the PDFs for the values of H0 based on the seven calibrations of

the TRGB discussed in §2. The width of each Gaussian is based on the statistical

uncertainties alone for each individual determination. The error bars at the top of

the plot (using the same color coding) represent the corresponding systematic uncer-

tainties in each case. The 1σ uncertainties are determined from the 16th and 84th

percentiles for the Frequentist sum of the distributions, adding the statistical and sys-


tematic errors in quadrature: σi =√σstat,i2 + σsys,i2, and σmean =

∑σi/√

(N − 1),

where N = 4. The four objects with independent geometric distances (the LMC,

Milky Way, NGC 4258 and the SMC) are represented by Gaussians with unit area.

The secondary calibrations of Sculptor, Fornax, and the LMC clusters are based on

the Milky Way calibration of the horizontal branch, and are therefore not completely

independent. Once again, their areas have been scaled following Equation 1 and are

shown for illustrative purposes only. Thus Sculptor, Fornax and the LMC clusters do

not contribute to the adopted overall calibration, but they do provide a consistency

check on the horizontal-branch-to-TRGB distance scale.

From Figure 7, it can also be seen that the range in the values of H0 for the

various calibrators is small relative to the published systematic error bars. The small

χ2 value may be indicating that the systematic errors have been over-estimated, or,

alternatively that statistical fluctuations have resulted in a fortuitously tight group-

ing of H0 values. In either case, a conservative estimate of the overall uncertainty

still seems warranted; that is, we do not consider this (better than 1% statistical)

agreement to be indicating that H0 has now been measured to a level of 1%.

In Figure 8, we show the normalized relative PDFs for the values of H0 based on

the different calibrators (LMC, NGC 4258, Milky Way, SMC), comparing both the

TRGB and Cepheid calibrations in a self-consistent manner. For comparison with

the SHoES results (where the statistical and systematic uncertainties are not treated

independently), only the total uncertainties are considered. The TRGB calibrations

are shown at the top (in red) and the Cepheid calibrations in the middle (in blue).

In this case, we follow a Bayesian approach, assuming that each anchor is equally

valid, and adopting a uniform prior. The bottom panel shows the product of the

PDFs. In the case of the Milky Way, the H0 values are based on the calibration

from Cerny et al. (2020) for the TRGB, and R21 for Cepheids. (The earlier Cepheid

results for the Milky Way based on HST/WFC3 scanning parallaxes (R16) resulted

in a much higher value of H0 = 76.18 ± 2.17 km s−1 Mpc−1.) The resulting values of

H0 for the TRGB and Cepheids, respectively, are shown as solid lines. The difference

between the TRGB calibration with H0 = 69.8±0.6 (stat)±1.6 (sys) km s−1 Mpc−1

(this paper) and the Cepheid calibration with H0 = 73.2 ± 1.3 km s−1 Mpc−1 (R21)

represents a 1.6σ tension between the TRGB and Cepheid calibrations.

The tension between the TRGB and Cepheid calibrations is perhaps not a serious

problem given that systematic uncertainties can be difficult to identify, and 2σ is

indicating generally good agreement, given those challenges. However, unlike the

tension between the early universe (CMB results) and the local value of H0, the true

distances to galaxies are fixed with unique values. Rather than signifying potential

new physics in the early universe, this “local” tension is unambiguously signaling that

26 Freedman

62 64 66 68 70 72 74 76Ho

Rela

tive

Pro

bab

ilit

y

Distribution of Ho Values for TRGB Anchors

LMC

Milky Way

NGC 4258

SMC

Sculptor

Fornax

LMC clusters

Ho = 69.8 ± 0.6 (stat) ± 1.6 (sys)

Figure 7. Probability density functions for the values of H0 based on the seven calibrationsdescribed in §2. The direct geometric calibrations for the LMC, the Milky Way, NGC 4258,and SMC are independent of each other. The H0 values for Sculptor, Fornax and four LMCclusters are based on the Milky Way calibration of the horizontal branch (and are thereforenot completely independent). They are consistent with the direct geometric calibrations,but they are not included in the final calibration.

the uncertainties in one or both distance scales (out to and including the SNe Ia) have

been underestimated.

4. RECENT INDEPENDENT CALIBRATIONS OF THE CEPHEID ZERO

POINT

4.1. Gaia EDR3 Calibration of the Leavitt Law

Gaia EDR3, as described in §2.2.1, also presents the opportunity to derive a

new zero-point calibration for Milky Way Cepheids (e.g., R21, Owens et al. 2021,

in prep., Breuval et al. 2021). (The R21 results were shown in the third panel of

Figure 8). We discuss below the Owens et al. Gaia EDR3-based calibration of a

multi-wavelength sample of field Cepheids, and compare these calibrations with the

sample of field Cepheids analyzed by R21.


H0 values for TRGB and Cepheids

LMC

N4258

Milky Way

SMC

LMC

N4258

Milky Way

66 68 70 72 74 76 78 80H0

Riess (2021)

This paper

Rela

tive

Pro

bab

ilit

yD

istr

ibu

tion

TRGB

Cepheids

TRGB Cepheids

Figure 8. A comparison of the calibrations for the TRGB method and Cepheids, as listedin Table 5. Upper two panels: Probability density functions are shown for the independentcalibrations for each method: the LMC (red), NGC 4258 (purple), the Milky Way (blue)and the SMC (orange), in the case of the TRGB; and the LMC, NGC 4258, and the MilkyWay in the case of Cepheids. Bottom panel: a comparison of the product of the probabilitydensity functions for the TRGB method and Cepheids based on the results from the upperpanels. The TRGB results are shown in red; Cepheid results are shown in blue. Notethat the relative weights of the TRGB and Cepheid distributions are determined, to a largeextent, by the differing uncertainties adopted for the Milky Way calibrations, where theCepheid result assumes a highly optimistic view of the current Gaia EDR3 calibration.

Owens et al. (2021, in prep.) have analyzed Gaia EDR3 data for 49 Milky Way

field Cepheids in an attempt to provide a multi-wavelength calibration of the Leavitt

law. In early anticipation of the Gaia mission Freedman et al. (2011) and Monson

et al. (2012) undertook a program to augment the sample of published optical pho-

tometry for Milky Way Cepheids with Spitzer mid-infrared (3.6 and 4.5 µm) photom-

28 Freedman

etry, providing a multiwavelength (BV RIJHK[3.6][4.5]) database for 37 Cepheids,

located both in the field and in open clusters.

Adopting the photogeometric distances obtained from the EDR3 parallax mea-

surements by Bailer-Jones et al. (2021), Owens et al. (2021, in prep.) derived optical-

to-mid-infrared Leavitt law relations for the Milky Way sample. The Bailer-Jones

et al. measurements include correction for the zero-point offset in Gaia EDR3 par-

allaxes (Lindegren et al. 2021a). A challenge at present is that this sample of Milky

Way Cepheids is very bright in apparent magnitude (4 < G < 11 mag). As already

discussed in §2.2.1, the corrected Gaia EDR3 parallaxes have large uncertainties, and

have been shown to be underestimates. Moreover, they are significantly underesti-

mated at brighter magnitudes (e.g., El-Badry et al. 2021), up to 30% for isolated

sources with small quoted astrometric uncertainties (and up to 80% for those with

companions). R21 found that a –14 µas correction to their Cepheid parallaxes was

indicated, obtained by minimizing the scatter in their Wesenheit Leavitt law.

In a comparison with HST parallaxes and published infrared Baade-Wesselink

distances, as well as the DEB distances to the LMC and SMC, Owens et al. (2021,

in prep.) concluded that the current uncertainty in their sample of EDR3 parallaxes

is conservatively at a level of ∼ ±5%, much larger than the 1% or better accuracy

anticipated from future (DR4 and DR5) Gaia releases. Owens et al. also explored

adding a constant offset to the Leavitt law, but found that there is no single offset

that minimizes the scatter (as would be expected for distance errors) for their mul-

tiwavelength sample. They instead used the DEB distances measured for the LMC

and SMC by Pietrzynski (2019) and Graczyk et al. (2020) to provide an external es-

timate of the offset in the Milky Way sample, finding a value of +17.5 µas, similar in

magnitude, but opposite in sign to that found by R21. (The sense of the offset found

by Owens et al. is in the same sense as that found by Maız Apellaniz et al. (2021).)

However, as Owens et al. emphasize, the adoption of the DEB distances does not

then provide an independent Gaia EDR3 zero-point calibration, and uncertainty in

the required correction to the Gaia EDR3 parallaxes remains.

Although the uncertainties are not yet at a level of 1%, there is still internal

consistency at a few percent level in the Cepheid zero points obtained using different

Cepheid samples, different parallax measurements, different external constraints and

analyzed by different authors. At this level, it provides evidence for stability in the

Cepheid zero point, much as we saw for the internal consistency and stability in the

TRGB zero point in §2. Once again, these results indicate that the divergence of the

TRGB and Cepheid distance scales, and the resulting values of H0, occur (at least

primarily) farther out in the rungs of the distance ladder, and are not coming from

errors in the respective locally determined zero-point calibrations.


5. COMPARISON OF THE TRGB AND CEPHEID CALIBRATIONS OF HO

The adopted TRGB value of H0 = 69.8 ± 0.6 (stat) ± 1.6 (sys) km s−1 Mpc−1

is smaller than the most recent SHoES Cepheid calibration at a level of ∼2σ. Next,

we examine the implications of forcing a higher value of H0 onto the calibration of

the TRGB for globular clusters in the Milky Way.

Figure 9 shows an expanded version of the MI versus (V-I)o color-magnitude

diagram for the Milky Way globular clusters discussed in §2.2, this time centered on

the giant branch. Corresponding values of H0 are indicated. It can be seen that a

value of H0 = 74 km s−1 Mpc−1 (R19) is significantly discrepant with the measured

position of the TRGB, as are the values of H0 = 75-76 km s−1 Mpc−1 calibrated

using scanning parallaxes for Milky Way Cepheids from R19, a recent calibration of

the Tully-Fisher relation Kourkchi et al. (2020), and surface-brightness fluctuations

(Verde et al. 2019). Values of H0 of 74 and 76 km s−1 Mpc−1 correspond to adopting

absolute magnitudes for the TRGB of MI = −3.92 and −3.86 mag, respectively,

significantly fainter than virtually all calibrations found in the published literature

(see Table 1), and differing by 6% and 9% from the calibration adopted here. Future

work is required to ascertain the reason for this discrepancy, most importantly, 1)

further comparisons of individual TRGB and Cepheid distances to SN Ia host galaxies,

and 2) the ultimate establishment of the zero point of both the TRGB and the Cepheid

distance scales at a <1% level with future Gaia releases. For comparison, the Planck

value of H0 would correspond to adopting values of MI = −4.12 mag (a 3% difference

from our adopted calibration).

In §2, §3, and §4, we have seen that recent updates to the absolute zero-points of

the TRGB and Cepheid distance scales are each internally consistent with previously

published zero points for each method at the 1–2% level, and therefore that the

difference in the values of H0 based on these two methods cannot be (completely)

ascribed to a zero-point error. A difference in H0 of 4 km s−1 Mpc−1 (i.e., between

70 and 74 km s−1 Mpc−1) corresponds to a difference of 0.12 mag or 6% in distance,

which is about 3-6 times the quoted uncertainty in the current estimates of the TRGB

and Cepheid zero points (of 1 to 2%).

As discussed in F19, the TRGB and Cepheid distances to galaxies agree well

(having a scatter of ±0.05 mag or 2% in distance) for nearby distances (< 7 Mpc),

but they begin to diverge for the more distant galaxies (where the scatter is over

three times larger, ±0.17 mag or 8% in distance), with a weighted average difference

in distance modulus (in the sense of TRGB minus Cepheid; i.e., the TRGB distances

are larger) amounting to +0.059 mag. Although in principle, one could adopt a

TRGB zero point that is significantly fainter than –4.05, that simply shifts the offset

to (and worsens the good agreement at) closer distances where the current Cepheid

30 Freedman

Figure 9. Composite MI versus (V − I)o color-magnitude diagram for giant branch stars,based on a sample of 46 Galactic globular clusters, color coded by the density of points.This plot is an expansion of the red rectangle shown in Figure 1. The TRGB is shown bythe red line, located at an absolute I-band magnitude of MI = -4.049 mag. This calibrationresults in a value of Ho = 69.8. Shown also for comparison as blue dashed lines are thecorresponding values for H0 = 67.4, 74 and 76, respectively. PARSEC(Padova and TriesteStellar Evolutionary Code: http://stev.oapd.inaf.it/cgi-bin/cmd) isochrones (CMD Version3.3; Bressan et al. (2012); Marigo et al. (2017)) with [Fe/H] values from left to right of -2.0,-1.2 and -0.8 dex, respectively, are illustrated by the three white curves outlined in black.The fits to these isochrones illustrates, both empirically and theoretically, how small theeffect of metallicity is for the TRGB in the I band at these low metallicities. The historicalH0 values of 100 and 50 are also labeled: their large spread relative to current measurementsillustrates the dramatic progress in the measurement of H0 in recent decades.

and TRGB distances agree extremely well, with an average difference of +0.02 mag

or 1% in distance.

The strengthening of both the TRGB and Cepheid zero-point calibrations, in

addition to the good agreement between the TRGB and Cepheids for distances closer

than 7 Mpc, again suggests that the discrepancy in H0 arises farther afield. One

potential clue as to part of the problem may be indicated by the observed scatter

in the calibrated absolute SNe Ia magnitudes, as discussed by F19. These authors

found that the scatter in the TRGB-calibrated SNe Ia magnitudes for nearby galaxies


amounted to σ = 0.11 mag, in good agreement with the scatter in the CSP Hubble

diagram of σ = 0.10 mag for the more distant SNe Ia sample, whereas the scatter

in the Cepheid-calibrated SNe Ia magnitudes is larger, with σ = 0.15 mag. Further

improvement to the distances of galaxies in the 15-30 Mpc range will be needed to

resolve this issue. Scheduled JWST observations will be critical to this effort (e.g.,

JWST Cycle 1 GO proposal(Proposal 01995; Freedman 2021).

In summary, the good agreement for the nearby sample suggests that the zero-

point calibrations of the methods are not the (primary) reason for the differences

between the two methods in determining H0. Resolving the reason for this divergence

is now critical to our understanding of whether there is new physics beyond the

standard ΛCDM model.

6. THE TRGB AND CEPHEIDS AS DISTANCE INDICATORS

Given the historical record of large and poorly understood disagreements amongst

various distance indicators (for example, the 50 versus 100 discrepancy illustrated in

Figure 9), the current (smaller) range of 67 to 74 in the value of H0 also reflects the

recent significant improvement in the extragalactic distance scale. That said, in the

context of testing the standard cosmological model, it is essential to understand the

origin of the difference in the TRGB and Cepheid distance scales.

We now turn to a discussion of each of the two methods individually. In specific,

we discuss the status of the calibrations, the viability of each method as a standard

candle, the effects of crowding/blending for each case, as well as the uncertainties due

to dust and metallicity. We highlight the particular strengths of each method, as well

as the current level of control of known systematic effects, and then outline prospects

for improvement.

6.1. Measuring TRGB Distances

1. Calibration of the TRGB zero point: As shown in §2 of this paper, direct geo-

metric calibrations of the TRGB method for the LMC (F19, F20; Hoyt (2021)),

NGC 4258 (Jang et al. 2021), globular clusters in the Milky Way (Freedman

et al. 2020; Cerny et al. 2020)), and the SMC (Hoyt 2021) all agree to within

±1%.

2. The TRGB as a Standard Candle: The strikingly sharp and flat definition of

the TRGB at F814W (comparable to the ground-based I band) for Milky Way

globular clusters (see Figure 9) provides growing direct evidence that old, blue

metal-poor giant branch stars at the tip of the RGB are actual standard candles,

distinctive from other commonly employed standardizable candles (for example,

32 Freedman

SNe Ia and Cepheids). The fact that this sharp cutoff is not simply an empirical

feature, but that it is the result of a well-understood physical mechanism (the

core helium flash) lends confidence to the use of these stars as reliable distance

indicators.

3. Photometric Errors Due to Crowding/Blending Effects: The TRGB method

is best applied in the outer halos of galaxies (e.g., see the discussion in Jang

et al. 2021, and references therein), where the surface brightness of the galaxy

is low, and the overlapping of stellar point spread functions is minimal. Crowd-

ing/blending effects are not currently a significant source of uncertainty for the

TRGB method if carefully applied to stars in the outer halos of galaxies.

4. Effects of Dust: Foreground and Internal: Foreground Milky Way reddening cor-

rections are obtained from the all-sky extinction maps of Schlafly & Finkbeiner

(2011). Beyond the Milky Way, for the application of the TRGB method tar-

geted in the halos of galaxies, the effects of dust are small (e.g., Menard et al.

2010). For the four current anchors (LMC, Milky Way, NGC 4258, and the

SMC), the local line-of-sight circumstances are different for each case, and ex-

tinction and reddening corrections have been investigated in detail on a case-

by-case basis as described in Jang et al. (2018); Cerny et al. (2020); Freedman

et al. (2020) and Hoyt (2021). For an individual anchor, the distance uncer-

tainty attributed to this correction contributes to its systematic uncertainty;

however, for the determination of H0 based on several anchors, it contributes

only to the overall statistical error, and not to the final systematic uncertainty.

5. Metallicity Effects: For red giant branch stars, there is a metallicity (and con-

comitant color) dependence of the luminosity that is both predicted by theory

and independently confirmed by observation (e.g., Freedman et al. 2020). A

significant advantage of the TRGB method is that it has long been known that

the color of a star on the red giant branch is a direct indicator of the metallicity

of the star (e.g., Da Costa & Armandroff 1990; Carretta & Bragaglia 1998).

Given a known (flat) TRGB slope in the I band, the corresponding slope of the

giant branch luminosity with color, at any other given wavelength, is not arbi-

trary: it is a priori mathematically defined for the other wavelengths (Madore

& Freedman 2020). Empirically, the slope and zero points of the V IJHK red

giant branch terminations determined for the LMC and SMC agree with those

measured for Milky Way globular clusters to within their 1-σ uncertainties (F20;

Cerny et al. 2020).

For the purposes of the I-band (F814W ) calibration presented in this paper, the

effects of metallicity are negligible, given that only the bluest (metal-poor) stars

enter the calibration, and that the flat (color-independent) nature of the TRGB

in this restricted color regime is well-established (see, for example, Figure 9

above, and Figure 6 of Jang et al. 2020).


6. Future Prospects for the TRGB Distance Scale:

a) Strengthening the Zero-Point Calibration: In the future Gaia DR4 will pro-

vide twice as many observations compared to EDR3, and a new full-scale astro-

metric solution with a decrease in both the random and systematic uncertainties

(Lindegren et al. 2021a) compared to those discussed in §2.2.1 for EDR3.

b) Increasing the Number of SNe Ia Calibrators:

(i) As new SNe Ia are detected in galaxies at distances ≤ 30 Mpc, HST obser-

vations of the halos of the host galaxies can provide I-band TRGB distances

with precisions of better than 2% for a modest investment in telescope time.

Unique in this regard is that the method can be applied to galaxies of all types,

including edge-on spiral, S0 and elliptical galaxies, thus both increasing the

number of calibrators and also mitigating potential systematics in the SN Ia

data.

(ii) A combination of ground- and space-based observations can further

strengthen the calibration of the TRGB at other (near- or mid-)infrared wave-

lengths (e.g., Dalcanton et al. 2012; Madore et al. 2018; Hoyt et al. 2018; Durbin

et al. 2020). Red giant stars are brighter in the near-infrared than at optical

wavelengths. With JWST , the mid-infrared TRGB calibration can be applied

to distances of '40 Mpc (or a volume five times greater than currently possible

with HST ), thereby adding significantly more SNe Ia into the calibration.10

6.2. Measuring Cepheid Distances

1. Calibration of the Cepheid Zero Point: Direct geometric calibrations of the

Cepheid Leavitt Law for the LMC are based on (a) the DEB distance to the

LMC (Pietrzynski 2019); (b) HST and Gaia parallaxes for field Cepheids in

the Milky Way (Benedict et al. 2007; Riess et al. 2018, 2021); and (c) the maser

distance to NGC 4258 (Reid et al. 2019). The resulting values of H0 for these

three calibration methods currently span a range of 72-74 km s−1 Mpc−1.11

2. Cepheids as Standardizable Candles: The well-defined relationship between pe-

riod, luminosity and color can, in principle, produce a standardizable candle

of high precision. Then, including a metallicity term, the Leavitt law can be

expressed as

Mλ1 = α logP + β(mλ1 −mλ2)o + γ[O/H] + δ (3)

where the Cepheid magnitude at a given wavelength λ1 is a function of the

logarithm of the period (P), a color term with coefficient β, and a term with

10 This new JWST capability is highly desirable because SNe Ia are sufficiently rare that host galaxiesfor which TRGB stars (or Cepheids) are also accessible with HST are discovered only every 1.5 to2 years.

11 Efstathiou (2020) has discussed at some length the internal tension between the LMC and NGC4258 anchor distances (which depend upon the adopted metallicity correction), and notes that theH0 tension may be arising, in part, due to inconsistencies in the local anchors.

34 Freedman

coefficient, γ, that allows for a metallicity effect (where [O/H] represents the log-

arithmic oxygen to hydrogen ratio for HII regions in the vicinity of the Cepheids,

relative to the solar value); and δ is the zero point.

It has long been recognized that the decreasing scatter in the correlation be-

tween period and luminosity with increasing wavelength (e.g., Madore & Freed-

man 1991), as well as the decreasing effect of reddening and metallicity with

increasing wavelength, motivates the application of the Leavitt Law at near-

infrared (or longer) wavelengths (McGonegal et al. 1982; Madore & Freedman

1991; Freedman et al. 1991; Macri et al. 2001; Freedman et al. 2008).12

3. Photometric Errors Due to Crowding/Blending Effects: Cepheid variables are

yellow supergiants, generally found in relatively high-surface-brightness areas

in the star-forming disks of late-type galaxies. For nearby galaxies, the crowd-

ing and blending of Cepheids is not a serious practical issue for the brightest,

long-period Cepheids, but the problem worsens as the distance increases and

the angular resolution decreases. Using artificial star tests, R16 concluded that

these crowding/blending effects do not induce systematic effects. In addition,

Riess et al. (2020) tried to infer the quantitative effects of crowding by com-

paring the amplitudes of Cepheids in four galaxies out to a distance of 20 Mpc.

They concluded that the erroneous measurements of Cepheid backgrounds alone

cannot explain the Hubble tension. Future work is still needed to assess the

implications for the even more distant galaxies in the SHoES program, which

extend out to 40-50 Mpc.

The effects of crowding/blending also become more severe with increasing wave-

length where, for a given aperture telescope, the resolution is poorer in the

infrared than in the optical. Disk red giants and the even-brighter asymptotic

giant branch (AGB) stars (both of which are redder than Cepheids) are the

main, and unavoidable, contaminants. Thus, although both dust and metal-

licity effects are decreasing functions of wavelength, there is a trade-off to be

made with the decreasing (wavelength-dependent) resolution, and the increas-

ing challenges of overlapping objects dominated by red stars, particularly as

the distance increases. In the case of HST and WFC3, the longest-wavelength

available, the F160W filter (comparable to the ground-based H band), has an

advantage for reducing the effects of dust and metallicity, but it is at a disad-

vantage in dealing with the effects of increased crowding and blending.

4. Effects of Dust : As a consequence of their relative youth, Cepheid variables

are unavoidably located close to the regions of dust and gas out of which they

formed. In practice, however, Cepheid reddening can be dealt with in a straight-

12 An exception is the 4.5µm band in which the Cepheid flux is affected by the presence of a CObandhead (Scowcroft et al. 2011).


forward manner. With accurate colors, Madore (1976, 1982) showed that a

reddening-free magnitude can be constructed; for example,

W = V −RV × (B − V ), (4)

where RV = AV /E(B-V) is the ratio of total-to-selective absorption. W has

been widely applied to the Cepheid distance scale (e.g., Freedman et al. 2001;

Riess et al. 2016). An advantage of W is that it simultaneously corrects for

all line-of-sight absorption, including both host-galaxy (internal) and Galactic

(foreground) reddening.13

5. Metallicity Effects: The effects of metallicity on the Cepheid Leavitt Law are

still being actively debated in the literature (e.g., for a recent summary see

Ripepi et al. 2020). One of the immediate challenges in constraining any metal-

licity effect for Cepheids is the difficulty of determining abundances for the in-

dividual Cepheids themselves. Spectroscopic abundances have been measured

for Cepheids in the Milky Way and LMC (e.g., Romaniello et al. 2008); how-

ever, more distant Cepheids are generally too faint to measure abundances from

spectroscopy.

Three decades of empirical tests for a Cepheid abundance effect (the measure-

ment of γ in Equation 3 (e.g., Freedman & Madore 1990; Kennicutt et al. 1998;

Romaniello et al. 2008; Fausnaugh et al. 2015; Riess et al. 2016; Ripepi et al.

2020; Breuval et al. 2021) have not yet led to a consensus view on the mag-

nitude of the effect or even its sign, or indeed, whether there is an effect at

any given wavelength. Most of these studies have had to rely on the use of

[O/H] abundances for nearby HII regions as a proxy for the Cepheid metallici-

ties, which cannot generally be measured directly. Theoretical models suggest

that the effect of metallicity will be smaller at longer wavelengths, but there

also remain significant differences in the predicted effects on both the slope

and intercept of the period-luminosity relations with wavelength (Bono et al.

2008a; Ripepi et al. 2020), even at the long wavelength of the K band (2.2

µm). Ripepi et al. find that the slope of the metallicity term ranges from -0.04

to -0.36 mag/dex for fundamental pulsators, and from +0.23 to -0.30 mag/dex

for overtone Cepheids. Recently, incorporating Gaia EDR3 data for the Milky

Way and comparing to the LMC and SMC, Breuval et al. (2021) find that the

metallicity effect is negligible in the optical (V band) and moreover, contrary

to previous studies, conclude that the effect increases through IJHK, with the

largest effect being in the near-infrared.

13 As noted recently by (Mortsell et al. 2021), however, if the assumption of a universal value for RV

is not valid, it could result in a systematic error in H0, an issue that could become increasinglyimportant in an era for which the goal is percent level accuracy.

36 Freedman

As we enter an era where 1-2% accuracies are required to resolve whether there is

an H0 tension, it is critical that the longstanding uncertainties due to metallicity

be better understood and calibrated. R16 compute a Wesenheit function of the

form:

MWH = mH −RH,V I × (V − I) where R ≡ AH/(AV − AI) (5)

and solve for a metallicity correction on a star-by-star basis. Their conclusion is

that metallicity contributes only at the 0.5% level to their total H0 uncertainty

of 2.4%. Given the long-standing disagreement in the literature (both from

theory and observations) further work is clearly warranted to confirm this as-

sertion. This issue is best addressed with multi-wavelength, high signal-to-noise

data for nearby galaxies where covariant crowding effects are less severe.

6. Summary and Future Prospects for the Cepheid Distance Scale: Cepheids have

many strengths that make them good distance indicators. However, they still

face a number of challenges, particularly when it comes to applying them under

conditions at the limits of current telescopes and detectors, with the goal of

achieving distances accurate and precise to a level of 1-2%. The main challenge

for Cepheid standardization is that several wavelengths, each of equally-high

precision, are required: first to correct for reddening; second to correct for a

possible metallicity effect (the wavelength dependence and sign of which remain

under debate); and third, to ensure that the effects of crowding/blending are

not systematically influencing the results.

All three of the above systematic effects (reddening, metallicity and crowd-

ing) increase toward the centers of galaxies. Since Cepheids are being

crowded/blended particularly by red giant and red (even brighter) AGB stars,

all three effects also will act in the sense of causing Cepheids to appear redder

in regions of coincidentally higher metallicity. Put another way, the corrections

for reddening, metallicity and crowding/blending are covariant; for example, if

the currently applied metallicity or crowding corrections are incorrect, then the

reddening corrections will also be in error, because they all involve the same

limited sets of colors, making it difficult to break the degeneracy. These issues

will continue to pose a serious challenge for 1% accuracy, especially when the

scatter in the observed Wesenheit Leavitt law can be 20-25% in distance or

0.4-0.5 mag in distance modulus (R16), even for (anchor) galaxies as close as

7.6 Mpc (e.g., NGC 4258).

There are many areas where future tests could further constrain uncertainties

in the Cepheid distance scale.


a) High signal-to-noise, multi-color, time-averaged (BV IJHK) photometry and

spectroscopy for nearby galaxies with a range of metallicities can help resolve

the question of the magnitude, sense and wavelength dependence of metallicity

corrections. The inclusion of additional distant galaxies will not lead to bet-

ter constraints on the systematic effects such as metallicity; obtaining larger

samples of galaxies will simply reduce the statistical uncertainties alone.

b) As further SNe Ia are discovered in the nearby universe, the numbers of

SNe Ia host galaxies with observable Cepheids will also slowly be increased.

c) JWST/NIRCam in the J band has four times the angular resolution of

HST/WFC3 in the H band, where the longest-wavelength SHoES Cepheid mea-

surements have been made, and thus can allow the effects of crowding/blending

in the HST photometry to be assessed directly.

6.3. Overall Systematics

At present, the systematic accuracies of the TRGB and the Cepheid distance-

scale zero-points are constrained by the small number of available geometric calibra-

tors providing high-accuracy distances. Below we outline the degree to which the

two distance scales are co-dependent (or not), on the same (or different) zero-point

calibrators.

As illustrated in Table 7, there are four galaxies with geometric measurements

that have been used to calibrate the local distance scale: the LMC, NGC 4258, the

Milky Way and the SMC. There are several important points to take away from this

table.

1. Both the TRGB and Cepheids adopt the same distances for the LMC and

NGC 4258, therefore sharing any systematic errors that may have been in-

curred in those measurements. The current total uncertainties quoted for these

measurements are at a level of 1% and 1.5%, respectively (Pietrzynski 2019;

Reid et al. 2019).

2. NGC 4258 is the only galaxy sufficiently nearby for which an accurate maser

distance can be measured, and which is also close enough for the calibration

of the TRGB and Cepheids. A “sample of one” precludes rigorous testing for

potential systematic errors in this galaxy’s geometric distance.

3. In the case of the TRGB method, the LMC and SMC calibrations share the

systematic uncertainties of the Skowron et al. (2021) reddening maps. The

dominant uncertainty is that of the zero point, estimated by Hoyt (2021) to be

± 0.014 mag (0.6%) and ± 0.018 mag (0.8%), respectively. In addition, their

38 Freedman

Table 7. Zero-Point Calibration

Calibrator TRGB Cepheids

LMC DEBa DEBa

NGC 4258 masersb masersb

Milky Way ω Cen DEBc EDR3 parallaxesd

SMC DEBe ...a Pietrzynski (2019)b Reid et al. (2019)c Thompson et al. (2001)d Riess et al. (2021)e Graczyk et al. (2020)

DEB distances are both based on the surface-brightness-color relation from

Pietrzynski (2019), estimated to be 0.8%.

4. Finally, it should be noted that for both the TRGB and Cepheids, the same

reddening law is adopted and assumed to be universal; moreover, the same ratio

of total-to-selective absorption, RV , is adopted in both applications. However,

the TRGB method is less susceptible to the assumption of the universality of

the reddening law because the dust content in the halos of galaxies is generally

neglible compared to that in the disks.

6.3.1. The SN Ia Host Galaxies

The tie-in to the more distant SN Ia host galaxies is similarly limited by the fact

that SNe Ia in the local universe are rare. As noted previously, there are currently

19 published Cepheid distances to SN Ia hosts, an equal number for which there

is a TRGB calibration, and a sample of 10 galaxies for which there is an overlap.

Any peculiarities in the SNe Ia in this overlap sample (that are not shared by the

more distant supernovae in the Hubble flow) will carry covariant systematics into the

TRGB and Cepheid H0 determinations.

Once again, the same reddening law is adopted for both the TRGB and Cepheids.

It is assumed to be universal, and the same value or RV is adopted in both applica-

tions. However, the explicit Galactic foreground reddening corrections used for the

TRGB are decoupled from the Cepheid de-reddening process that implicitly corrects

for total line-of-sight reddening using the Wesenheit method.


6.3.2. Distant SNe Ia in the Hubble Flow

Both the TRGB and Cepheids tie in to more distant SNe Ia in the Hubble flow

for the final step in the determination of H0. While different filters, different software

analysis tools, and different groups have analyzed the data, any unknown systematics

in SN Ia distances will be shared by both methods. However, the TRGB method,

which can be applied to both elliptical and spiral galaxies, will be less sensitive to

correlations that are host-galaxy mass dependent.

As the distances to more SN Ia host galaxies are measured using HST and JWST ,

the statistical (uncorrelated) errors will decrease as 1 /√

(N). As the above discussion

makes clear, however, there are parts of the systematic error budgets for the TRGB

and Cepheid H0 determinations that are covariant. Unfortunately, quantifying these

potential covariant effects (many of which fall into the category of current unknowns:

e.g., reddening laws, unknown systematics in masers, DEBs, SNe Ia etc.) is not a

realistic prospect. Ultimately, completely independent methods (e.g., gravitational

wave sirens) will be required to test for and place external constraints on covariant

systematics in the local distance scale.

7. COMPARISON WITH OTHER RECENT DETERMINATIONS OF HO

To date, only ten SNe Ia host galaxies have both TRGB and Cepheid distances

measured (F19). Yet this sample is significantly larger than available for any other

primary distance indicator. Stated another way, this is the first independent and

direct test for individual galaxies in the Cepheid-supernova distance scale, and signif-

icant differences between the TRGB and Cepheid distances have been found. What

about other tests?

Although a case has been made that there are many independent checks of the

Cepheid distance scale (e.g., Verde et al. 2019), the small number of galaxies currently

available precludes a detailed and direct comparison of the Cepheid distance scale with

most other distance indicators. For example, the Mira method (Huang et al. 2018)

is currently based upon the detection of these long-period variable stars in a single

galaxy, NGC 4258, calibrated via masers in that galaxy. Furthermore, the calibration

of H0 using this method then relies on observations of a single SN Ia host galaxy, NGC

1559 (Huang et al. 2020).

Similarly, NGC 4258, at a distance of 7.6 Mpc, is the only galaxy in the nearby

universe where the host is close enough to have a measured Cepheid distance where

the maser technique can also be applied (Reid et al. 2019). Furthermore, there are

only six galaxies in total (including NGC 4258) for which maser distances have been

measured and used to estimate H0, (Pesce et al. 2020); the statistical errors for this

40 Freedman

technique are thus still large compared with, for example, SNe Ia (SNe Ia; Scolnic

et al. 2018; Burns et al. 2018) where samples of a hundred or hundreds of SNe Ia

have been measured. The five additional megamaser galaxies beyond NGC 4258

have distances ranging from 50 to 130 Mpc (with recession velocities of 680 to 10200

km/s), and peculiar-velocity corrections remain a significant source of uncertainty.

(An average peculiar velocity correction of 250 km/s is about 30% of the recession

velocity of 679 km/s for NGC 4258.) For the total sample of six galaxies, Pesce

et al. find values of H0 ranging from 71.8 to 76.9 km s−1 Mpc−1, depending on what

assumptions are made, and/or which models are adopted for the peculiar velocities.

In Figure 10, we show a comparison of several recent determinations of H0 and

their published uncertainties. Plotted are the relative probability density functions

color-coded as labeled in the legend and include: the TRGB (this paper); Cepheids

(R21); those based on early-universe measurements (CMB: Planck Collaboration et al.

(2020), the Dark Energy Survey Year 3 + BAO + BBN (DES Collaboration et al.

2021); as well as gravitational wave sirens (Hotokezaka et al. 2019); Miras (Huang

et al. 2020); surface brightness fluctuations (SBF Khetan et al. 2021; Blakeslee et al.

2021); masers (Reid et al. 2019); and recent results from strong lensing (Birrer et al.

2020). The Planck, DES Year 3 + BAO + BBN, TRGB and Cepheid PDFs are also

explicitly labeled.

From this figure, the discrepancy between the early universe (CMB + BAO) and

local Cepheid measurements of H0 is apparent, as is the difference between the TRGB

and Cepheid local determinations. Both the TRGB and Cepheid measurements have

smaller uncertainties than the other (local) methods shown. These two methods

currently have the largest samples of nearby objects (19 in both cases) that tie directly

into the Hubble flow via SNe Ia.

Thus, the current situation is that there are two different types of tensions in

play: 1) that between Cepheid measurements and the early universe and 2) that

between Cepheid measurements and the TRGB.

For completeness, in Appendix A, we show a plot of 1,065 H0 values as a function

of time for published data since 1980 (Ian Steer, private communication), as well as

their histogram distribution. Interestingly, there is no bimodality (67 versus 73) seen

in the overall distribution of the recently published H0 values, as can be seen in

Figure A2.

8. SUMMARY

In this paper, we have provided an update on the calibration of the absolute

I-band magnitude of the TRGB anchored using several independent geometric zero


64 66 68 70 72 74 76 78H0

Rela

tive

Pro

bab

ilit

yD

en

sity

Recent Published H0 Values

Planck

TRGB

Cepheids

Lensing

DES+BAO+BBN

GW Sirens

Miras

SBF

Masers

SN II

Planck

DES Y3+BAO+BBN

TRGB

Cepheids

Figure 10. Relative probability density functions for several current methods for measur-ing H0. The CMB, BAO, strong lensing and TRGB methods currently yield lower values ofH0, while Cepheids yield the highest values. The uncertainties associated with H0 measure-ments from gravitational wave sirens, strong lensing, Miras, masers, and SBF are currentlysignificantly larger than the errors quoted for the TRGB and Cepheids. See text for details.(CMB: Planck Collaboration 2018; TRGB: this paper; Cepheids: R21; Lensing: Birrer et al.(2020); DES Y3 + BAO + BBN: DES Collaboration et al. (2021); GW sirens: Hotokezakaet al. (2019) Miras: Huang et al. (2018); SBF: Khetan et al. (2021); Masers: Reid et al.(2019)).

points. This updated calibration includes 1) extensive measurements of the TRGB

over a wide area in the halo of the maser galaxy NGC 4258 (Jang et al. 2021); 2)

independent observations of the TRGB in 46 Milky Way globular clusters covering a

wide range of metallicities (Cerny et al. 2020); and 3) a reanalysis of the TRGB in-

corporating revised reddening corrections for the LMC and SMC (Hoyt 2021). These

calibrations all agree with that earlier determined for the LMC alone (F19, F20) to

better than 1%, providing multiple consistency checks on the LMC calibration of F19

and F20. Each of these calibrations is tied to geometrical distance anchors (H2O

megamasers in the case of NGC 4258; DEB distances and Gaia EDR3 parallaxes for

the Milky Way globular clusters; and DEB distances for the LMC and the SMC). In

addition, using a fiducial horizontal branch sequence defined by the Milky Way glob-

ular clusters, we discuss and compare the TRGB absolute magnitude for the nearby

dwarf elliptical galaxies Sculptor (Tran et al. 2021, in prep) and Fornax (Oakes et

al. 2021, in prep), and for four LMC globular clusters, finding excellent additional

agreement.

42 Freedman

An improved value of H0 is determined by applying this new TRGB calibration

to a sample of distant SNe Ia. This measurement is based on: 1) the new calibration

of the absolute I-band magnitude of the TRGB (MTRGBF814W = −4.049 ± 0.015 (stat) ±

0.035 (sys) mag) presented in this paper; 2) HST/ACS observations of TRGB stars

in the halos of nearby galaxies known to host SNe Ia (F19, F20, Hoyt et al. (2021));

3) a sample of 99 well-observed SNe Ia with multiwavelength photometry from the

CSP (Krisciunas et al. 2017). Our final adopted value is

Ho = 69.8± 0.6 (stat)± 1.6 (sys) kms−1Mpc−1. (6)

This value of H0, based on the TRGB, agrees to within 1.3σ with that inferred from

modeling of the CMB observations.

Currently the TRGB method and Cepheids provide the largest (statistically ro-

bust) and strongest (tested for systematics) base of distance determinations for the

calibration of H0 in the local universe. Together they provide a check on the overall

systematics. It is a testament to each method that a comparison for the nearest

galaxies (i.e., within 7 Mpc) agrees in both zero point and scatter to better than 2%

accuracy (F19). However, these same two distance scales diverge at larger distances.

It is important to understand the source of this divergence, and to ascertain whether

its resolution will strengthen or weaken the case for additional physics. The fact that

any given galaxy must have a unique distance means that systematic errors in one or

both of the current estimates must be the cause for the divergence. At this time, the

outcome is unknown: no clear evidence for outstanding systematic effects in either

the TRGB or Cepheid distances has been found. It should be noted, however, that

crowding/blending effects are not an issue for the TRGB, that multiple geometric de-

terminations of the zero point show consistency at the 1% level, and that metallicity

effects are better understood from theory, and more easily addressed empirically, for

TRGB stars than for Cepheids. Finally, a number of ongoing studies of the TRGB

and Cepheids, combined with the upcoming launch of JWST , plus improvements to

the Gaia zero points in future releases, all hold promise for significant improvement

leading to a resolution of the current discrepancies within the next few years.

ACKNOWLEDGMENTS

Support for program #13691 was provided by NASA through a grant from the

Space Telescope Science Institute, which is operated by the Association of Universities

for Research in Astronomy, Inc., under NASA contract NASA 5-26555. The CSP-I

has been supported by the National Science Foundation under grants AST0306969,

AST0607438, AST1008343, AST1613426, and AST1613472. Computing resources


2000 2005 2010 2015 2020 2025 2030Year of Publication

60

65

70

75

80H

0[k

ms−

1M

pc−

1]

Hubble Constant Over Time

Cepheids CMB ( Planck, ACT+W) TRGB

Cepheids

TRGB

CMB

Figure 11. A summary of Hubble constant values in the past two decades, based onCepheid variables (blue squares), the TRGB (red filled circles and star), and estimatesbased on measurements of fluctuations in the CMB (WMAP: black filled diamonds; Planck:yellow diamonds; ACT + WMAP: cyan diamond). The CMB H0 values assume a flat ΛCDM model. The CMB and Cepheid results straddle a range of 67 to 74 km s−1 Mpc−1,with the TRGB results falling in the middle, and overlapping the CMB results. The tensionbetween the CMB and TRGB results amounts to only 1.3σ.

used for this work were made possible by a grant from the Ahmanson Foundation.

This research has made use of the NASA/IPAC Extragalactic Database (NED), which

is operated by the Jet Propulsion Laboratory, California Institute of Technology,

under contract with the National Aeronautics and Space Administration. Some of

the data presented in this paper were obtained from the Mikulski Archive for Space

Telescopes (MAST). STScI is operated by the Association of Universities for Research

in Astronomy, Inc., under NASA contract NAS5-26555. I thank the Observatories of

the Carnegie Institution for Science and the University of Chicago for their support

of long-term research into the calibration and determination of the expansion rate

of the Universe. My thanks to many collaborators who have contributed to various

facets of this research on the TRGB, Cepheids and supernovae; in particular Barry

Madore for his many decades of collaboration on the distance scale; as well as current

and previous students Taylor Hoyt, William Cerny, Quang Tran, Elias Oakes, Kayla

Owens, Finian Ashmead, and Dylan Hatt; postdoctoral fellows In Sung Jang, Rachael

Beaton and Jill Neeley; research scientists and faculty Andy Monson, Mark Phillips,

Mark Seibert, Jeff Rich and Myung Gyoon Lee. Special thanks to Chris Burns for

re-running his SNooPy and STAN MCMC code on the CCHP TRGB sample updated

since 2020, and for creating Figure 6. In addition, I gratefully acknowledge Ian Steer

for providing access to his H0 database. I thank Barry Madore, Kayla Owens, In

Sung Jang and Taylor Hoyt for their comments on the manuscript, as well as an

anonymous referee for several helpful suggestions to update and improve the paper.

44 Freedman

Facilities: HST (ACS) Gaia

Software: Matplotlib, NumPy, SciPy

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1980 1985 1990 1995 2000 2005 2010 2015 2020Year of Publication

20

40

60

80

100

120H

0[k

ms−

1M

pc−

1]

Published Hubble Constants

Figure A1. Plot of published H0 values since 1980. Data courtesy of Ian Steer, pri-vate communication. These data provide an update of the John Huchra Hubble constantdatabase originally maintained for the NASA Hubble Space Telescope Key Project on theextragalactic distance scale (Freedman et al. 2001). This figure further updates that shownin Steer (2020) with an additional 99 entries.

APPENDIX

A. HUBBLE CONSTANTS PUBLISHED SINCE 1980

Figure A1 plots H0 values published since 1980. The scatter in published H0

values has continued to decrease with time. All methods are included, without judge-

ment as to accuracy of a given method. In this sense it is an unbiased sample.

Figure A2 shows in histogram form the distribution of H0 values. It illustrates

clearly how the scatter in H0 values has decreased over the past four decades. For the

most recent decade (2010-2020), the average, median and mode of the H0 distribution

are 68.9, 68.6 and 68.0 km s−1 Mpc−1, respectively. The values of H0 inferred from

measurements of the CMB are shown in black. Interestingly, no obvious bimodality

48 Freedman

Figure A2. Histogram distributions of H0 values for all published data since 1980 (yellow),data since 2000 (purple), data since 2010 (cyan) and Ho estimates from CMB data (black).Data source is the same as for Figure A1.

of H0 values is seen between the values of 67 and 74 km s−1 Mpc−1, the two values

that define the current “H0 tension”.

arXiv:2106.15656v1 [astro-ph.CO] 29 Jun 2021

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